OORBITAL STABILITY OF INTERNAL WAVES
ROBIN MING CHEN AND SAMUEL WALSH
Abstract.
This paper studies the nonlinear stability of capillary-gravity waves propagating alongthe interface dividing two immiscible fluid layers of finite depth. The motion in both regions isgoverned by the incompressible and irrotational Euler equations, with the density of each fluid beingconstant but distinct. A diverse collection of small-amplitude solitary wave solutions for this systemhave been constructed by several authors in the case of strong surface tension (as measured bythe Bond number) and slightly subcritical Froude number. We prove that all of these waves are(conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shownto be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameterregion. For the near critical surface tension regime, we prove that one can infer conditional orbitalstability or orbital instability of small-amplitude traveling waves solutions to the full Euler systemfrom considerations of a dispersive PDE model equation.These results are obtained by reformulating the problem as an infinite-dimensional Hamiltoniansystem, then applying a version of the Grillakis–Shatah–Strauss method recently introduced in[51]. A key part of the analysis consists of computing the spectrum of the linearized augmentedHamiltonian at a shear flow or small-amplitude wave. For this, we generalize an idea used by Mielke[45] to treat capillary-gravity water waves beneath vacuum.
Contents
1. Introduction 12. Hamiltonian formulation for internal waves 73. Spectral analysis 184. Proof of the main results 33Acknowledgments 36Appendix A. Elementary identities 36References 381.
Introduction
We consider the classical problem of determining the evolution of a free boundary dividing twosuperposed incompressible, inviscid, and immiscible fluids under the influence of gravity. Thissituation arises in countless applications, with a particularly important example being internal wavespropagating along a pycnocline or thermocline in the ocean. Recent years have seen enormousprogress made in understanding the Cauchy problem for this system, and there is now a robust(local) well-posedness theory. In parallel, a large body of work has established the existence ofmyriad traveling wave solutions. Far less is known about the stability of these waves. While manyauthors have addressed the spectral or linear stability of interfacial waves, nonlinear results aremostly limited to dispersive model equations such as Kortweg–de Vries (KdV). In this paper, weprove a number of theorems on the (conditional) orbital stability of small-amplitude traveling wavesolutions to the full system when the surface tension is strong in a sense to be quantified shortly.
Date : March 1, 2021. a r X i v : . [ m a t h . A P ] F e b R. M. CHEN AND S. WALSH y = η ( t, x ) d + d − Ω + ( t )Ω − ( t ) Figure 1.
Configuration of the internal wave system. The unshaded fluid region Ω + ( t ) has density ρ + while the darker shaded region Ω − ( t ) below is of density ρ − ≥ ρ + . Their interface S ( t ) is a free boundary given by the graph of η = η ( t, x ) .In the far field, the widths of the upper and lower layer limit to d + and d − , respectively.Mathematically, the problem is formulated as follows. Fix Cartesian coordinates ( x, y ) so that thewave propagates in the x -direction with gravity acting in the negative y -direction. Because we aremost interested in the motion of the boundary, we suppose that the fluid domain is confined to achannel with rigid walls at heights y = ± d ± for fixed d ± ∈ (0 , ∞ ) . At each time t ≥ , the interface S = S ( t ) is taken to be the graph of an unknown smooth function η = η ( t, x ) . For small-amplitudewaves, this choice incurs no loss of generality. Then, the upper layer inhabits the (time-dependent)set Ω + = Ω + ( t ) := (cid:8) ( x, y ) ∈ R : η ( t, x ) < y < d + (cid:9) , while the lower layer is given by Ω − = Ω − ( t ) := (cid:8) ( x, y ) ∈ R : − d − < y < η ( t, x ) (cid:9) . We write Ω( t ) := Ω + ( t ) ∪ Ω − ( t ) to denote the fluid domain. Our focus will be on spatially localizedwaves for which η ( t, · ) decays at infinity. See Figure 1 for an illustration.Assuming that the flow in each region is irrotational and incompressible, the velocity field in Ω ± ( t ) is then given by ∇ Φ ± , for some function Φ ± = Φ ± ( t, x ) called the velocity potential. We takethe density in Ω ± ( t ) to be constant and denote it by ρ ± > . In order to ensure that heavier fluidelements do not lie above lighter elements, it is required that ρ + ≤ ρ − . The case ρ + = 0 formallycorresponds to a single fluid beneath vacuum. All of our analysis extends to this regime with onlysuperficial modifications to the arguments.The evolution of the system is governed by the incompressible irrotational Euler equations with afree boundary. In the bulk, the conservation of momentum has the simple expression(1.1a) ∆Φ ± = 0 in Ω ± ( t ) . On both the rigid and moving boundary components, we have the kinematic condition(1.1b) (cid:40) ∂ t η = − η (cid:48) ∂ x Φ − + ∂ y Φ − = − η (cid:48) ∂ x Φ + + ∂ y Φ + on { y = η ( t, x ) } ∂ y Φ ± = 0 on { y = ± d ± } , while on S ( t ) the dynamic or Bernoulli condition is imposed:(1.1c) (cid:115) ρ∂ t Φ + 12 ρ |∇ Φ | + gρη (cid:123) + σ (cid:32) η (cid:48) (cid:112) η (cid:48) ) (cid:33) (cid:48) = 0 on { y = η ( t, x ) } . Here (cid:74) · (cid:75) := ( · ) + − ( · ) − denotes the jump of a quantity over the interface, g > is the gravitationalconstant, and σ > is the coefficient of surface tension. The last term on the right-hand side aboveis the signed curvature of the interface and represents the influence of capillary effects. In (1.1b), weare enforcing the continuity of the normal velocity across the interface, while (1.1c) arises from theYoung–Laplace law for the pressure jump. Also, here and in what follows we will mostly adhere to NTERNAL WAVE STABILITY 3 βλ β Γ Γ Γ • λ ABC
Figure 2.
Bifurcation diagram for internal capillary-gravity waves. Region A isthe lighter shaded area that lies above Γ and to the right of Γ ; this is where onehas monotone solitary waves. Region B consists of all ( β, λ ) lying above Γ and tothe right of Γ . Finally, Region C is the darker shaded set neighboring Γ . Explicitparameterizations for these curves can be found in (2.34) and (2.44)the convention that primes denote x -derivatives of functions depending on ( t, x ) , while ∂ x is reservedfor functions of ( t, x, y ) or in defining operators.Rather than work with the full velocity potential Φ ± , which is defined on a moving domain, it isadvantageous to consider its restriction to the free boundary: ϕ ± = ϕ ± ( t, x ) := Φ ± ( t, x, η ( t, x )) . Through the use of nonlocal operators, it is possible to reformulate (1.1) in terms of the surfacevariables ( η, ϕ + , ϕ − ) ; see Section 2.1.1.1. Informal statement of results.
Traveling or steady solutions of (1.1) are waves of permanentconfiguration that appear independent of time when viewed in a moving reference frame. Specifically,they exhibit the ansatz η ( t, x ) = η c ( x − ct ) , ϕ ± ( t, x ) = ϕ c ± ( x − ct ) , for some traveling wave profile ( η c , ϕ c + , ϕ c − ) and wave speed c ∈ R . In the gravity wave case σ = 0 ,it is known that there exist solitary waves [11, 4, 44, 33], for which η c decays as | x | → ∞ ; periodicwaves [4, 5], for which η c is periodic in x ; and fronts [5, 43, 44, 19, 20], for which η c has distinct limitsupstream and downstream. Without surface tension, however, the dynamical problem is ill-posed[38], so to study stability we always take σ > . Rigorous existence results for small-amplitudeperiodic waves (including those with vorticity) were obtained in this regime by Le [40]. Solitaryinternal capillary-gravity waves were constructed by Kirrmann [36] and Nilsson [46]; the stability ofthese solutions is the main subject of the present paper. We also note that analytical and numericalinvestigations of this regime have been performed by Laget and Dias [37].The existence and qualitative properties of traveling internal waves are determined by fourdimensionless parameters. The primary two are the Bond number β and inverse square Froudenumber λ given by(1.2) β := σd + ρ − c , λ := − g (cid:74) ρ (cid:75) d + ρ − c . R. M. CHEN AND S. WALSH
The Bond number measures the strength of the surface tension, while λ describes the balance betweenkinetic and potential energy. One can think of the Froude number / √ λ as a non-dimensionalizedwave speed, hence large λ corresponds roughly to slow moving waves.The dispersion relation for internal capillary-gravity waves (rescaled to dimensionless variables) isgiven by(1.3) (cid:88) ± ρ ± ρ − ξ coth (cid:18) d ± d + ξ (cid:19) = λ + βξ . This results from linearizing the problem at the trivial solution ( η, ϕ + , ϕ − ) = (0 , , , then looking foreigenvalues of the form iξ . If ξ is a root to (1.3), the linearized problem admits a plane wave solutionwith η = exp ( iξ ( x − ct )) . After some algebra, it can be shown that there are three bifurcationcurves Γ , Γ , Γ that organize the ( β, λ ) -plane into regions where the configuration of the spectrumnear the imaginary axis is qualitatively the same; see Figure 2. They meet at the point ( β , λ ) ,which is given by(1.4) β := 13 (cid:18) ρ + ρ − + d − d + (cid:19) , λ := ρ + ρ − + d + d − , and there we find that ξ = 0 is a root of (1.3) with multiplicity . We say that β is the criticalBond number separating the weak and strong surface tension regimes.In this regard, the internal wave system is quite similar to that of water waves beneath vacuum;see, for instance, [2, 31, 24, 32, 15, 16, 29]. However, there are two additional parameters to consider:the ratios of the fluid densities (cid:37) and far-field layer heights h , defined by(1.5) (cid:37) := ρ + ρ − , h := d − d + . These are specific to the two-fluid problem and allow for a surprisingly rich variety of traveling waves.For example, it has been proved by Nilsson [46] and Kirrmann [36] that for ( β, λ ) in the Region Aillustrated in Figure 2, there exists six qualitatively distinct types of small-amplitude waves. When (cid:37) − /h is negative and O (1) as λ (cid:38) λ , they find waves of depression (that is, η < ) that are toleading order KdV solitons. These are the only kind of wave possible in the corresponding parameterregime for the one-fluid case, which is consistent with simply taking ρ + = 0 . On the other hand,when (cid:37) − /h > , there are internal waves of elevation ( η > ) whose interface is a perturbedKdV soliton. Moreover, in the regime | (cid:37) − /h | (cid:104) | λ − λ | / (cid:28) , they construct traveling wavesthat are Gardner solitons to leading order. This furnishes four types of solutions, with waves ofdepression and elevation for both signs of (cid:37) − /h . A fuller account is given in Section 2.5.Our first theorem, stated informally for the time being, establishes the nonlinear stability of allthese waves in the orbital sense. Theorem 1.1 (Strong surface tension) . Every sufficiently small-amplitude solitary internal wave ( η c , ϕ c + , ϕ c − ) with ( β, λ ) in Region A and < λ − λ (cid:28) is conditionally orbitally stable in thefollowing sense. For all R > and r > , there exists r > such that, if ( η, ϕ + , ϕ − ) is any solutiondefined on a time interval [0 , t ) that obeys the bound (1.6) sup t ∈ [0 ,t ) (cid:16) (cid:107) η ( t ) (cid:107) H + (cid:107) ϕ + ( t ) (cid:107) ˙ H
52 + ∩ ˙ H + (cid:107) ϕ − ( t ) (cid:107) ˙ H
52 + ∩ ˙ H (cid:17) < R, and for which the initial data satisfies (1.7) (cid:107) η (0) − η c (cid:107) H + (cid:107) ϕ + (0) − ϕ c + (cid:107) ˙ H + (cid:107) ϕ − (0) − ϕ c − (cid:107) ˙ H < r , then (1.8) sup t ∈ [0 ,t ) inf s ∈ R (cid:16) (cid:107) η ( t, · − s ) − η c (cid:107) H + (cid:107) ϕ + ( t, · − s ) − ϕ c + (cid:107) ˙ H + (cid:107) ϕ − ( t, · − s ) − ϕ c − (cid:107) ˙ H (cid:17) < r. NTERNAL WAVE STABILITY 5
Remark . The bound in (1.8) controls the distance between ( η, ϕ + , ϕ − ) and the family of translatesof the steady wave. This is natural given that the underlying system (1.1) is translation invariant,and indeed it is necessary even for model equations such as KdV. Local well-posedness for the Cauchyproblem at the level of regularity represented by the norm in (1.6) has been proved by Shatah andZeng [49, 50]. On the other hand, we will show in Section 2.3 that the lower regularity norm in (1.7)and (1.8) is equivalent to the physical energy. We also emphasize that because r is independent of t , this result is much stronger than continuity of the data-to-solution map. For a global-in-timesolution, it gives orbital stability in the classical sense.Our next result concerns uniform flows for which the interface is perfectly flat and the velocity ispurely horizontal with the same constant value c in both layers. In a reference frame moving withthe wave, it therefore appears quiescent. While linear stability criteria for this regime are classical(see, for example, [25]), as far as we are aware, this is the first nonlinear stability result. Theorem 1.3 (Uniform flow) . The laminar solution ( η c , ϕ c + , ϕ c − ) = (0 , , is conditionally stablein the sense of Theorem 1.1 provided that ( β, λ ) lies in Region B. Lastly, we consider the critical surface tension case where ( β, λ ) lies in Region C near ( β , λ ) ; seeFigure 2. It is well-established that in this regime, the dynamics of sufficiently shallow waves arecaptured by a fifth-order nonlinear dispersive PDE similar to the Kawahara equation [35, 10]. Forspatially localized traveling waves, one can then integrate to obtain a fourth-order ODE(1.9) Z (cid:48)(cid:48)(cid:48)(cid:48) − δ ) Z (cid:48)(cid:48) + Z − Z = 0 , where we have scaled out all but the non-dimensional parameter δ = δ c , which is determined explicitlyby the wave speed via (2.41). The ODE (1.9) boasts an extraordinarily large variety of solutionsthat are homoclinic to (see, for example, [16]). For this paper, we focus on the family { Z δ } of“primary homoclinic” orbits that are even, unimodal, and exponentially localized. They have beenrigorously constructed for δ ≥ and − (cid:28) δ < , and numerically observed to persist as δ (cid:38) − .Nilsson [46] shows that for every | δ c | (cid:28) , there exists a traveling wave ( η c , ϕ c + , ϕ c − ) solution to(1.1) with η c given to leading order by a rescaling of Z δ c . The next result states that the orbitalstability or instability of these solutions to the full internal wave problem can be determined byconsiderations of the far simpler model equation (1.9). Theorem 1.4 (Critical surface tension) . Let { Z δ } be the family of primary homoclinic solutions to (1.9) and suppose that ( β, λ ) lies in Region C with | δ c | (cid:28) . Then the corresponding traveling wavesolution ( η c ∗ , ϕ c ∗ + , ϕ c ∗ − ) to (1.1) is conditionally orbitally stable provided that the function (1.10) c (cid:55)→ sgn c (cid:90) R Z δ c d x is strictly increasing at c ∗ , and it is orbitally unstable if this function is strictly decreasing there. We remark that this theorem is new even for the one-fluid case. Physically, the integral in (1.10)represents the momentum carried by the wave; whether it is increasing or decreasing as a function of δ has been investigated by many authors but remains open in the present case. Under conditionsanalogous to Theorem 1.4, Levandosky [41, 42] proves a nonlinear stability/instability result forground state solutions to a family of fifth-order dispersive PDEs that includes the Kawahara equation.On the other hand, for δ = 1 / , the primary homoclinic solution to (1.9) has the explicit formula Z = 3524 sech (cid:32) √ · (cid:33) , and by exploiting this, (1.10) can be evaluated directly for various choices of the dimensionalparameters [1, 23, 34]. Numerical evidence in [30] suggests that stability holds for the Kawaharaequation with δ > , but analytical results are not currently available. Through Theorem 1.4,progress on this question for the model equation can immediately be translated to (1.1). R. M. CHEN AND S. WALSH
Idea of the proof.
It is well known that the internal wave problem (1.1) can be formulatedas an abstract Hamiltonian system of the general form ∂ t u = J D E ( u ) , where u = u ( t ) is an unknown related to ( η, ϕ + , ϕ − ) , the Poisson map J is a skew-adjoint operator,and E is a conserved energy functional. The translation invariance of the system gives rise to asecond conserved quantity, the momentum P . A traveling wave solution with wave speed c is in facta critical points of the augmented Hamiltonian E c := E − cP .It is therefore natural to adopt a constrained variational viewpoint, attempting to show that thewaves are minimizers of the energy on level sets of the momentum. A serious challenge that arises inmany applications, including the present one, is that D E c has an unstable direction as well as a eigenvalue due to translation invariance. This situation can lead to either stability or instability,and a deft use of the conserved quantities is necessary to discern which occurs for the waves inquestion. Benjamin [7] pioneered this approach in his study of the orbital stability of KdV solitons.A systematic and greatly expanded version was later developed by Grillakis, Shatah, and Strauss[26]. Now called the GSS method, it is one of the primary tools in nonlinear stability theory forHamiltonian systems.Historically, though, GSS has not been especially successful in treating the full water waveproblem. Indeed, (1.1) exhibits a host of features that make it highly resistant to naïve applicationsof systematic methods. For example, the theory in [26] requires that J be an isomorphism, whichdoes not hold here as we show in Section 2.3. It is also formulated under the hypothesis that theCauchy problem is globally well-posed in the natural energy space. At present, (1.1) is only knownto be locally well-posed and this assumes considerably more smoothness. Because the water waveproblem is quasilinear, it is not expected to generate a flow on the energy space. Worse still, thecorresponding functional E is not even differentiable at this level or regularity.Seeking to address these issues, Varholm, Wahlén, and Walsh [51] obtained a variant of the GSSmethod that weakens the above hypotheses. In place of the bijectivity of J , it essentially requiresonly that J is injective with dense range. The functional analytic framework is also designed toaccommodate the gap in regularity between the energy space and the smoothness needed for localwell-posedness. In this paper, we use the relaxed GSS method to attack the water wave problemdirectly and prove Theorem 1.1 and Theorem 1.4. A simpler, self-contained argument suffices forTheorem 1.3 as the augmented linearized Hamiltonian has no unstable directions in that case.The most challenging step in this procedure is computing the spectrum of the linearized augmentedHamiltonian at a traveling wave. For this, we generalize a technique introduced by Mielke [45] inhis work on solitary capillary-gravity waves in a single finite-depth fluid and with strong surfacetension. Briefly, this involves using the kinematic condition to eliminate ϕ ± and obtain an auxiliaryfunctional acting only on η . Conjugating by a rescaling operator, a delicate argument shows that forsufficiently small-amplitude waves, the spectrum coincides to leading order with the linearizationof a dispersive model equation (steady KdV or Gardner in the setting of Theorem 1.1 and steadyKawahara for Theorem 1.4). Here it is important to note that these calculations are substantiallymore difficult in the internal wave setting than for a single fluid: the nonlocal operators introducedin the Hamiltonian reformulation are more complicated, and they must be expanded to higher order.On the other hand, Mielke proves conditional orbital stability using an ad hoc modification of theGSS method. Because we have at our disposal the general theory from [51], we are able to streamlinethis part of the argument.Let us also mention an alternative variational approach to proving nonlinear stability of waterwaves due to Buffoni. Roughly speaking, this consists of a penalization scheme followed by aconcentration compactness argument to directly construct traveling waves as constrained minimizersof the energy with fixed momentum. In some circumstances, one can then apply a soft analysisargument of Cazenave and Lions [17] to infer so-called (conditional) energetic stability , meaning that NTERNAL WAVE STABILITY 7 the set of constrained minimizers is stable in the energy norm. This differs from the orbital stabilitywe obtain unless one also has uniqueness of the minimizer up to translation, which is typicallynot available. Through this variational method, Buffoni proved the existence and stability (in theabove sense) of solitary waves in the single-fluid case with strong surface tension [12]. He also gavepartial results concerning waves with weak surface tension and in infinite depth [13, 14]. Pushingsignificantly further the technique, Groves and Wahlén [27, 28] subsequently obtained completeversions of these theorems, and also treated the case of constant vorticity [29].1.3.
Plan of the article.
In Section 2, we begin by reformulating the internal wave problem (1.1)as an abstract Hamiltonian system in the style of Benjamin and Bridges [8]. A number of hypothesesnecessary to apply the general theory in [51] are then be verified. We also recall the existence theorydue to Nilsson [46], recasting it within the Hamiltonian framework of the present paper.Section 3 is devoted to computing the spectrum of the linearized augmented Hamiltonian at auniform flow or small-amplitude traveling wave. As mentioned above, our calculation is patterned onthe basic approach of Mielke [45], but with many additional challenges owing to the more complicatedphysical setting.The main results are then proved in Section 4. Thanks to the general theory, this requires usonly to determine whether the so-called moment of instability, a scalar-valued function of the wavespeed, is strictly convex or concave. This is accomplished by exploiting a long-wave rescaling andthe leading-order form of the waves known from the existence theory.Finally, Appendix A contains some elementary calculations that plan an essential part in thespectral computation.2.
Hamiltonian formulation for internal waves
Nonlocal operators and surface variables.
Following the classical Zakharov–Craig–Sulemidea, we will reformulate the interface Euler equations (1.1) as a nonlocal problem in terms ofquantities restricted to the free boundary S ( t ) . A similar approach was taken by Benjamin andBridges [8] and Craig and Groves [21] in their treatments of this system.Recall that we have defined ϕ ± ( t, x ) := Φ ± ( t, x, η ( t, x )) , to be the traces of the velocity potentials for the upper and lower regions. The velocity field canthen be recovered by means of the Dirichlet–Neumann operator in Ω ± ( t ) . For a fixed η , this is themapping given by G ± ( η ) f ± := (cid:104) η (cid:48) (cid:105) (cid:0) N ± · ∇H ± ( η ) f (cid:1) | S (2.1)where N ± is the unit outward normal to Ω ± along S , we are making use of the Japanese bracketnotation (cid:104) · (cid:105) := (cid:112) | · | , and H ± ( η ) f is the harmonic extension of f to Ω ± . Specifically, in viewof the kinematic conditions (1.1b) on the rigid boundaries, we take H ± ( η ) f to be the unique solutionto(2.2) ∆ H ± ( η ) f = 0 in Ω ± H ± ( η ) f = f on { y = η } ∂ y H ± ( η ) f = 0 on { y = ± d ± } . Dirichlet–Neumann operators are a standard tool in the study of water waves; for a general reference,see [39] or [48]. In particular, for any real numbers k > / and k ∈ [1 / − k , / k ] , and profile η ∈ H k +1 / ( R ) with − d − < η < d + , we have that G ± ( η ) is an isomorphism ˙ H k ( R ) → ˙ H k − ( R ) ,where ˙ H k denotes the usual homogeneous Sobolev space of order k . Similarly, H ± ( η ) is bounded asa mapping H k ( R ) → H k +1 / (Ω ± ) and ˙ H k ( R ) → ˙ H k +1 / (Ω ± ) . Our analysis relies on the fact thatthe Dirichlet–Neumann operator depends smoothly on η . Indeed, η (cid:55)→ G ± ( η ) is real analytic and at R. M. CHEN AND S. WALSH η = 0 , it is the Fourier multiplier G ± (0) = | ∂ x | tanh ( d ± | ∂ x | ) . Note also that G ± ( η ) is self-adjoint ˙ H / ( R ) → ˙ H − / ( R ) and positive definite.Because N + + N − = 0 , the continuity of the normal velocity over the interface is equivalent to(2.3) G + ( η ) ϕ + + G − ( η ) ϕ − = 0 . Thus the kinematic condition (1.1b) on S ( t ) can be expressed as(2.4) ∂ t η = ∓ G ± ( η ) ϕ ± . Note that the kinematic condition on { y = ± d ± } is encoded in the definition of H .Rather than work with ϕ ± , we consider the quantity(2.5) ψ := − (cid:74) ρ Φ (cid:75) = ρ − ϕ − − ρ + ϕ + . Using (2.3), we can recover both ϕ + and ϕ − from ψ . Indeed, we compute that − G − ( η ) ψ = ρ + G − ( η ) ϕ + − ρ − G − ( η ) ϕ − = ρ + G − ( η ) ϕ + + ρ − G + ( η ) ϕ + = B ( η ) ϕ + , where(2.6) B ( η ) := ρ + G − ( η ) + ρ − G + ( η ) . By the above discussion, we have that B ( η ) is bounded and linear H k ( R ) → H k − ( R ) and ˙ H k ( R ) → ˙ H k − ( R ) , for all η ∈ H k +1 / ( R ) and with k , k given as before. One can readily confirm, moreover,that B ( η ) is an isomorphism ˙ H k ( R ) → ˙ H k − ( R ) . Thus, repeating the same computation with signsreversed leads to the identity(2.7) ϕ ± = ∓ B ( η ) − G ∓ ( η ) ψ. The kinematic condition (2.4) can then be recast as(2.8) ∂ t η = A ( η ) ψ for the operator(2.9) A ( η ) := G − ( η ) B ( η ) − G + ( η ) . It is simple to show that these operators commute, and hence we can alternatively write A ( η ) = G + ( η ) B ( η ) − G − ( η ) . To reformulate the Bernoulli condition (1.1c) requires being able to reconstruct the full gradient ∇ Φ ± restricted to the interface from the surface variables. For this, we simply observe that ϕ (cid:48)± = ( ∂ x Φ ± ) | y = η + η (cid:48) ( ∂ y Φ ± ) | y = η , which together with the definition of G ± ( η ) in (2.1) leads to the useful identities(2.10) (cid:18) ϕ (cid:48)± G ± ( η ) ϕ ± (cid:19) = (cid:18) η (cid:48) ± η (cid:48) ∓ (cid:19) ( ∇ Φ ± ) | S , ( ∇ Φ ± ) | S = 11 + ( η (cid:48) ) (cid:18) ± η (cid:48) η (cid:48) ∓ (cid:19) (cid:18) ϕ (cid:48)± G ± ( η ) ϕ ± (cid:19) . Now, observe that simply by definition − ∂ t ψ = ρ + ∂ t ϕ + − ρ − ∂ t ϕ − = (cid:74) ρ∂ t Φ (cid:75) + ( ∂ t η ) (cid:74) ρ∂ y Φ (cid:75) . Thus (1.1c) can be rewritten as(2.11) ∂ t ψ = 12 (cid:113) ρ |∇ Φ | (cid:121) − ( ∂ t η ) (cid:74) ρ∂ y Φ (cid:75) + g (cid:74) ρ (cid:75) η + σ (cid:18) η (cid:48) (cid:104) η (cid:48) (cid:105) (cid:19) (cid:48) . In view of (2.10), this gives a formulation of the Bernoulli condition involving only the surfacevariables η and ψ . NTERNAL WAVE STABILITY 9
Functional analytic setting.
Let us now define the function spaces in which the internalwave problem will be posed. Following the approach outlined above, we wish to recast the systemin terms of the unknown u := ( η, ψ ) . It is convenient to introduce a scale of spaces describing thespatial regularity of u : for each k ≥ / , let(2.12) X k = X k × X k := H k + ( R ) × (cid:16) ˙ H k ( R ) ∩ ˙ H ( R ) (cid:17) . In what follows, we will frequently use the shorthand X k + (and likewise H k + ) to denote X k + ε forany < ε (cid:28) that is fixed and then suppressed. Remark . Observe that H r ( R ) ∩ ˙ H s ( R ) is dense in both H r ( R ) and ˙ H s ( R ) for all r, s ∈ R ; see,for example, [51, Lemma A.1].We will work in a trio of nested Banach spaces W (cid:44) → V (cid:44) → X . The largest, X , we call the energyspace . Specifically, we take(2.13) X := X = H ( R ) × ˙ H ( R ) . Its dual is X ∗ = H − ( R ) × ˙ H − ( R ) , and we let I = (1 − ∂ x , | ∂ x | ) denote the natural isomorphism X → X ∗ . In particular, when u ∈ X ,the velocity field ∇ Φ ± ∈ L (Ω ± ) . As we will see below, this ensures that the kinetic energy isindeed finite. Likewise, the H norm of η is equivalent to the excess potential energy relative to theundisturbed state.However, observe that u (cid:55)→ G ± ( η ) is not smooth with domain X , since we must have that η is atleast Lipschitz continuous and also bounded away from the rigid boundaries at y = ± d ± . This leadsus to introduce the space(2.14) V := X = H + ( R ) × (cid:16) ˙ H ( R ) ∩ ˙ H ( R ) (cid:17) , and neighborhood O := { ( η, ψ ) ∈ X : − d − < η < d + } . Note that H / ( R ) (cid:44) → W , ∞ ( R ) , so u ∈ V does indeed imply that η has the requisite Lipschitzcontinuity.Lastly, because the Cauchy problem is not likely to be well-posed in V , we consider the evensmoother space(2.15) W := X + = H ( R ) × (cid:16) ˙ H + ( R ) ∩ ˙ H ( R ) (cid:17) . Local well-posedness at this level of regularity was proved by Shatah and Zeng [50], for example.Before continuing, we record the fact these spaces have the following embedding property thatcorresponds to [51, Assumption 1].
Lemma 2.2 (Spaces) . Let the spaces X , V , and W be defined by (2.13) , (2.14) , and (2.15) , respec-tively. There exists a constant C > and θ ∈ (0 , ) such that (cid:107) u (cid:107) V ≤ C (cid:107) u (cid:107) θ X (cid:107) u (cid:107) − θ W for all u ∈ W . Proof.
This can be quickly verified using from the definitions of X , V , and W and the Gagliardo–Nirenberg interpolation inequality. (cid:3) Observe that this inequality ensures that small cubic terms in V are dominated by quadratic termsin X on bounded sets in W , which is needed in the general theory when Taylor expanding functionalsthat are smooth with domain V ∩ O . A similar argument appears in the proof of Theorem 4.1. Hamiltonian structure.
Benjamin and Bridges [8] established that the internal wave problem(1.1) has a (canonical) Hamiltonian formulation in terms of the state variable u by adapting thewell-known Zakharov–Craig–Sulem formulation for the single-fluid case. In this section, we will recallthe system obtained in [8] while verifying that it satisfies a number of the hypotheses of the generaltheory.The kinetic energy carried by the wave is given by K = 12 (cid:90) Ω + ( t ) ρ + |∇ Φ + | d x d y + 12 (cid:90) Ω − ( t ) ρ − |∇ Φ − | d x d y = 12 (cid:90) R ρ + ϕ + G + ( η ) ϕ + d x + 12 (cid:90) R ρ − ϕ − G − ( η ) ϕ − d x. Using (2.3) and (2.7), this can be rewritten as K = 12 (cid:90) R ψG − ( η ) B ( η ) − G + ( η ) ψ d x. Thus, we can view K as the C ∞ ( O ∩ V , R ) functional acting on u given by(2.16) K ( u ) := 12 (cid:90) R ψA ( η ) ψ d x, where recall A ( η ) was defined in (2.9). Likewise, the potential energy for the system is described bythe functional V ( u ) := − (cid:90) R g (cid:74) ρ (cid:75) η d x + σ (cid:90) R (cid:16)(cid:112) η (cid:48) ) − (cid:17) d x. The total energy is thus E ( u ) := K ( u ) + V ( u )= 12 (cid:90) R ψA ( η ) ψ d x − (cid:90) R g (cid:74) ρ (cid:75) η d x + σ (cid:90) R (cid:16)(cid:112) η (cid:48) ) − (cid:17) d x. (2.17)By our choice of spaces, E ∈ C ∞ ( O ∩ V ; R ) . We claim, moreover, that D E ( u ) can be extended toa mapping defined on the entire dual space X ∗ . This rather technical fact is necessary in order toreformulate the problem as a Hamiltonian system.Before addressing this question, we pause to record the following crucial formulas for the Fréchetderivatives of the nonlocal operators G ± ( η ) and A ( η ) . Lemma 2.3 (First derivatives) . Let ( η, ψ ) ∈ O ∩ V , ˙ η ∈ V , and ξ ∈ V be given. (a) The Fréchet derivative of G ± ( η ) admits the representation formula (cid:90) R ξ (cid:104) D G ± ( η ) ˙ η, ψ (cid:105) d x = (cid:90) R (cid:0) a ± ( η, ψ ) ξ (cid:48) + a ± ( η, ψ ) G ± ( η ) ξ (cid:1) ˙ η d x, (2.18) with a ± ( η, ψ ) := 11 + ( η (cid:48) ) (cid:0) ∓ ψ (cid:48) − η (cid:48) G ± ( η ) ψ (cid:1) a ± ( η, ψ ) := 11 + ( η (cid:48) ) (cid:0) ± G ± ( η ) ψ − η (cid:48) ψ (cid:48) (cid:1) . (2.19)(b) The Fréchet derivative of A ( η ) admits the representation formula (cid:90) R ξ (cid:104) D A ( η ) ˙ η, ψ (cid:105) d x = (cid:88) ± ρ ± (cid:90) R (cid:16) a ± ( η, A ( η ) G ± ( η ) − ψ ) (cid:0) A ( η ) G ± ( η ) − ξ (cid:1) (cid:48) (cid:17) ˙ η d x + (cid:88) ± ρ ± (cid:90) R (cid:0) a ± ( η, A ( η ) G ± ( η ) − ψ ) A ( η ) ξ (cid:1) ˙ η d x. (2.20) NTERNAL WAVE STABILITY 11
Remark . Observe that by (2.10), a ± ( η, ψ ) = ∓ ( ∂ x H ± ( η ) ψ ) | S while a ± ( η, ψ ) = − ( ∂ y H ± ( η ) ψ ) | S .In particular, this means that both are linear in ψ . Proof of Lemma 2.3.
The formula (2.18) for D G ± ( η ) can be derived using the same method as thestandard one-fluid case. To obtain (2.20), it is easier to first consider the derivative of(2.21) A ( η ) − = G + ( η ) − B ( η ) G − ( η ) − = ρ + G + ( η ) − + ρ − G − ( η ) − . Then, (cid:104) D A ( η ) ˙ η, ψ (cid:105) = − A ( η ) (cid:10) D ( A ( η ) − ) ˙ η, A ( η ) ψ (cid:11) = (cid:88) ± ρ ± A ( η ) G ± ( η ) − (cid:10) D G ± ( η ) ˙ η, G ± ( η ) − A ( η ) ψ (cid:11) . Using the self-adjointness of G ± ( η ) and the formula (2.18) for D G ± ( η ) , this leads immediately to(2.20). (cid:3) We are now able to prove that D E ( u ) extends to X ∗ when the base point u has sufficient regularity. Lemma 2.5 (Energy extension) . There exists a mapping ∇ E ∈ C ∞ ( O ∩ V ; X ∗ ) such that (cid:104)∇ E ( u ) , v (cid:105) X ∗ × X = D E ( u ) v for all u ∈ O ∩ V , v ∈ V . Proof.
Let u = ( η, ψ ) ∈ O ∩ V and ˙ u = ( ˙ η, ˙ ψ ) ∈ V be given. Then from the definition of E in (2.17)and the self-adjointness of A ( η ) , we compute thatD E ( u ) ˙ u = 12 (cid:90) R ψ (cid:104) D A ( η ) ˙ η, ψ (cid:105) d x + (cid:90) R ˙ ψA ( η ) ψ d x − (cid:90) R (cid:18) g (cid:74) ρ (cid:75) η + σ (cid:18) η (cid:48) (cid:104) η (cid:48) (cid:105) (cid:19) (cid:48) (cid:19) ˙ η d x. The latter two terms on the right-hand side certainly correspond to an element of X ∗ acting on ˙ u .To see the same is true for the first term, we make use of the representation formula (2.20) to write (cid:90) R ψ (cid:104) D A ( η ) ˙ η, ψ (cid:105) d x = (cid:88) ± ρ ± (cid:90) R a ± ( η, θ ± ) θ (cid:48)± ˙ η d x + (cid:88) ± ρ ± (cid:90) R (cid:0) a ± ( η, θ ± ) A ( η ) ψ (cid:1) ˙ η d x, for a ± and a ± given by (2.19) and θ ± := A ( η ) G ± ( η ) − ψ . Since u ∈ O ∩ V , it is easy to check that A ( η ) ψ, a ± ( η, θ ± ) , a ± ( η, θ ± ) ∈ L ( R ) , θ ± ∈ H ( R ) , and hence the extension ∇ E ( u ) can be defined explicitly as (cid:104)∇ E ( u ) , v (cid:105) X ∗ × X = ( E (cid:48) ( u ) , v ) L , where the L gradient E (cid:48) ( u ) = ( E (cid:48) η ( u ) , E (cid:48) ψ ( u )) takes the form E (cid:48) η ( u ) := 12 (cid:88) ± ρ ± (cid:0) a ± ( η, θ ± ) θ (cid:48)± + a ± ( η, θ ± ) A ( η ) ψ (cid:1) − g (cid:74) ρ (cid:75) η − σ (cid:18) η (cid:48) (cid:104) η (cid:48) (cid:105) (cid:19) (cid:48) ,E (cid:48) ψ ( u ) := A ( η ) ψ. (2.22)This completes the proof. (cid:3) Remark . Throughout the paper, we use the notational convention that, for a C functional F ( V ; R ) and u ∈ V , D F ( u ) ∈ V ∗ is the Fréchet derivative at u , F (cid:48) ( u ) is the L gradient, and ∇ F ( u ) is an extension of D F ( u ) to X ∗ (should such an extension exist).The energy space X will be endowed with symplectic structure through the prescription of thePoisson map(2.23) J := (cid:18) − (cid:19) : Dom J ⊂ X ∗ → X with domain(2.24) Dom J := (cid:16) H − ( R ) ∩ ˙ H ( R ) (cid:17) × (cid:16) H ( R ) ∩ ˙ H − ( R ) (cid:17) . While J appears relatively anodyne at first glance, the difference in regularity and homogeneitybetween X and X means that it is not bijective. This unpleasant fact is one of the major barriersto applying the classical GSS method [26] to the system. The next lemma shows, however, that J satisfies the weaker requirements of [51, Assumption 2]. Lemma 2.7 (Poisson map) . The Poisson map J defined by (2.23) satisfies the following. (a) Dom J is dense in X ∗ ; (b) J is injective; and (c) J is skew-adjoint in the sense that (cid:104) J u, v (cid:105) X ∗ × X = −(cid:104) u, J v (cid:105) X ∗∗ × X ∗ for all u, v ∈ Dom J. Proof.
Part (a) is a consequence of Remark 2.1, while (b) and (c) are obvious by definition. (cid:3)
Theorem 2.8 (Hamiltonian formulation) . Consider the abstract Hamiltonian system (2.25) ∂ t u = J D E ( u ) , u | t =0 = u where u ∈ O ∩ W is the initial data, J is the canonical symplectic matrix (2.23) , and the energy E is defined in (2.17) . We say u ∈ C ([0 , t ); O ∩ W ) is a (weak) solution to (2.25) provided dd t (cid:104) u ( t ) , w (cid:105) = −(cid:104)∇ E ( u ( t )) , J w (cid:105) for all w ∈ Dom J in the distributional sense on the time interval t ∈ [0 , t ) . This holds if and only if the corresponding ( η, Φ ± ) solves the Eulerian internal wave problem (1.1) .Proof. At this formulation of the problem was previously obtained by Benjamin and Bridges[8], we provide a sketch of the argument for completeness. Suppose that u ( t ) = ( η ( t ) , ψ ( t )) ∈ C ([0 , t ); O ∩ W ) is a weak solution to the Hamiltonian system (2.25). Recalling (2.7), we have that Φ ± := ∓H ( η ) G ± ( η ) − A ( η ) ψ ∈ ˙ H ∩ ˙ H is the velocity potential in Ω ± and satisfies (1.1a). Thedefinition of the harmonic extension operator H ( η ) in (2.2) also ensures the kinematic conditionholds on { y = ± d ± } . Moreover, from the expression for E (cid:48) ( u ) obtained in (2.22), we see that ∂ t η = E (cid:48) ψ ( u ) = A ( η ) ψ, in the distributional sense. This is precisely (2.8) and hence corresponds to the kinematic conditionon the internal interface (1.1b).We claim that the Bernoulli condition (1.1c) is equivalent to ∂ t ψ = − E (cid:48) η ( u ) . interpreted again in the distributional sense. Observe that, due to Remark 2.4 and the identity (2.7),many of the quantities occurring in E (cid:48) η ( u ) have physical significance: θ ± = A ( η ) G ± ( η ) − ψ = ∓ ϕ ± , a ± ( η, θ ± ) = ( ∂ x Φ ± ) | S , a ± ( η, θ ± ) = ± ( ∂ y Φ ± ) | S . Hence, E (cid:48) η ( u ) = 12 (cid:88) ± ρ ± (cid:0) ∓ ( ∂ x Φ ± ) | S ϕ (cid:48)± ± ( ∂ y Φ ± ) | S A ( η ) ψ (cid:1) − g (cid:74) ρ (cid:75) η − σ (cid:18) η (cid:48) (cid:104) η (cid:48) (cid:105) (cid:19) (cid:48) = − (cid:113) ρ |∇ Φ | (cid:121) + ( ∂ t η ) (cid:74) ρ∂ y Φ (cid:75) − g (cid:74) ρ (cid:75) η − σ (cid:18) η (cid:48) (cid:104) η (cid:48) (cid:105) (cid:19) (cid:48) , where in the second line we have used the kinematic condition (2.8) and the identities (2.10).Comparing this to equivalent statement of the Bernoulli condition in (2.11), we see that the proof isindeed complete. (cid:3) NTERNAL WAVE STABILITY 13
The symmetry group and the momentum.
The internal wave problem is invariant undertranslations in the x -direction, which formally should be associated to the conservation of (horizontallinear) momentum; see, for example, [9]. To put this on firmer ground, we introduce the one-parametersymmetry group(2.26) T ( s ) u := u ( · − s ) for all u ∈ X . In the next lemma, we verify that T exhibits the necessary properties for the abstract theory in [51]. Lemma 2.9 (Symmetry) . The translation symmetry group T given by (2.26) satisfies the following. (a) The neighborhood O , X k for any k , and I − Dom J are invariant under T ( s ) for all s ∈ R . (b) T comprises a flow on X in the sense that T (0) = Id X and T ( s + r ) = T ( s ) T ( r ) for all s, r ∈ R . Moreover, T ( s ) is unitary on X and an isometry on V and W for all s ∈ R . (c) The symmetry group commutes with the Poisson map in the sense that (2.27)
J IT ( · ) = T ( · ) J I. (d)
The infinitesimal generator of T | X k is the unbounded linear operator (2.28) T (cid:48) (0) | X k : Dom T (cid:48) (0) ⊂ X k → X k u (cid:55)→ − ∂ x u with (dense) domain Dom T (cid:48) (0) | X k := X k +1 . In particular,
Dom T (cid:48) (0) = Dom T (cid:48) (0) | X = X , Dom T (cid:48) (0) | V = X + , Dom T (cid:48) (0) | W = X + . (e) The subspace
Rng J ∩ Dom T (cid:48) (0) | W is dense in X . (f) We have E ( T ( s ) u ) = E ( u ) for all s ∈ R and u ∈ O ∩ V .Proof. Most of these facts are simple to confirm, so we omit the details. However, part (e) meritscloser consideration since its conclusion is the key assumption in [51] that replaces the hypothesisthat J is bijective in the standard GSS approach. First note that Rng J = (cid:16) H ( R ) ∩ ˙ H − ( R ) (cid:17) × (cid:16) H − ( R ) ∩ ˙ H ( R ) (cid:17) , and hence by part (d) we have that Dom T (cid:48) (0) | W ∩ Rng J = (cid:16) H ( R ) ∩ ˙ H − ( R ) (cid:17) × (cid:16) H − ( R ) ∩ ˙ H ( R ) ∩ ˙ H + ( R ) (cid:17) . This is indeed dense in X due to Remark 2.1. (cid:3) Now, letting P ± := ± (cid:90) R ρ ± η (cid:48) ϕ ± d x represent the momentum in Ω ± , we have that the total momentum carried by the wave is(2.29) P ( u ) := P + ( u ) + P − ( u ) = − (cid:90) R η (cid:48) ψ d x, which defines a C ∞ ( O ∩ V ; R ) functional. The next lemma establishes that P is indeed generated bythe translation invariance in the sense that (2.30) holds. In particular, together with Lemmas 2.5and 2.9, this completes the proof that [51, Assumption 3 and Assumption 4] hold. Lemma 2.10 (Momentum) . The momentum functional P given by (2.29) satisfies the following. (a) There exists a mapping ∇ P ∈ C ( O ∩ V ; X ∗ ) such that, for all u ∈ O ∩ V , ∇ P ( u ) is anextensions of the Fréchet derivative D P ( u ) . (b) For all such u ∈ O ∩ V it holds that ∇ P ( u ) ∈ Dom J and, moreover, (2.30) T (cid:48) (0) u = J ∇ P ( u ) for all u ∈ O ∩ Dom T (cid:48) (0) . Proof.
The existence of the extension ∇ P in part (a) is obvious from the formulas for the derivativeD P . In particular, for u = ( η, ψ ) ∈ O ∩ V and ˙ u = ( ˙ η, ˙ ψ ) ∈ V , we have(2.31) D P ( u ) ˙ u = (cid:90) R ψ (cid:48) ˙ η d x − (cid:90) R η (cid:48) ˙ ψ d x =: (cid:104)∇ P ( u ) , ˙ u (cid:105) X ∗ × X . The right-hand side above clearly defines an element of X ∗ that depends continuously on u . Inparticular, it has the explicit L gradient(2.32) P (cid:48) ( u ) = ( P (cid:48) η ( u ) , P (cid:48) ψ ( u )) , P (cid:48) η ( u ) := ψ (cid:48) , P (cid:48) ψ ( u ) := − η (cid:48) . From this it is also clear that ∇ P ( u ) ∈ Dom J for u ∈ O ∩ V . Noting that Dom T (cid:48) (0) = X / ⊂ V ,the identity (2.30) now follows from the definitions of J in (2.23) and T (cid:48) (0) in (2.28). (cid:3) Traveling waves.
In Hamiltonian language, a traveling internal wave is a solution to (2.25)taking the form(2.33) u ( t ) = T ( ct ) U, for some wave speed c ∈ R and time-independent bound state U ∈ O ∩ W . Let us now discuss insomewhat finer detail the existence theory obtained by Nilsson in [46].Recall that we have defined the dimensionless parameters β , λ , (cid:37) , and h in (1.2) and (1.5). Let T := { z ∈ C : Re z ∈ ( − r, r ) } be a thin slab centered on the imaginary axis. For r > sufficientlysmall, we have by the dispersion relation (1.3) that there exist three curves in the ( β, λ ) -plane alongwhich the spectrum of the linearized problem in T crosses the real or imaginary axis.Consider first the curve Γ , which is simply the line λ = λ . Immediately below it and to theright of β = β , the spectrum in T consists of a pair of oppositely signed real eigenvalues and acomplex conjugate pair on the imaginary axis. Passing through Γ , the imaginary eigenvalues collideat the origin then move along the real axis. This same resonance is associated with transitionfrom periodic solutions to solitons in the steady KdV equation, for example. On the curve Γ := { ( β ( ξ ) , λ ( ξ )) : ξ ∈ [0 , ∞ ) } , where β ( ξ ) := (cid:88) ± ρ ± ρ − d + d ± (cid:32) − sin ( d ± d + ξ ) cos ( d ± d + ξ ) + d ± d + ξ ξ sin ( d ± d + ξ ) (cid:33) ,λ ( ξ ) := β ( ξ ) + ξ (cid:88) ± ρ ± ρ − coth ( d + d ± ξ ) , (2.34)the spectrum in T consists of two real eigenvalues with multiplicity . In the region bounded by Γ and Γ , there are two pairs of oppositely signed simple real eigenvalues.Nilsson’s approach is to fix β > β and treat λ as a bifurcation parameter with < λ − λ (cid:28) .This ensures that ( β, λ ) remains in the Region A depicted in Figure 2, which is the narrow open setbounded below by Γ and lying beneath Γ . Because he opts to non-dimensionalize the system atthe outset, translating his result to our setting involves introducing some heavy notation. Thankfully,this will be pared down soon. Theorem 2.11 (Nilsson [46]) . Let { Π ε = ( ρ ± ε , d ± ε , σ ε , c ε ) : 0 < ε (cid:28) } be a smooth curve inthe dimensional parameter space such that the corresponding Bond number is fixed to β > β and λ = λ + ε . (a) Suppose that (cid:37) ε − /h ε = O (1) as ε (cid:38) . Then for any k > / , there exists a smooth curve C A β = { u A ε ; β : 0 < ε (cid:28) } ⊂ X k NTERNAL WAVE STABILITY 15 so that u A ε ; β is a traveling internal wave for the parameter values Π ε . Along this curve, thefree surface profile has leading-order form (2.35) η A ε ; β = ε d + (cid:37) − /h sech (cid:18) ε · d + √ β − β (cid:19) + O ( ε ) in X k as ε (cid:38) . (b) Suppose instead that (cid:37) ε − /h ε = κε for a fixed κ (cid:54) = 0 . Then for any k > / there existstwo smooth curves C A β,κ, ± = { u A ε ; β,κ, ± : 0 < ε (cid:28) } ⊂ X k so that u A ε ; β,κ, ± is a traveling internal wave for the parameter values Π ε . Along C A β,κ, ± , thefree surface profile has leading-order form (2.36) η A ε ; β,κ, ± = 2 εd + κ ± (cid:112) κ + 4( (cid:37) + 1 /h ) cosh (cid:18) ε · d + √ β − β (cid:19) + O ( ε ) in X k as ε (cid:38) .Remark . The above solutions are obtained using a center manifold reduction at the point ( β, λ ) ∈ Γ . For the scaling regime of part (a), the reduced equation is a perturbation of steady KdV.This gives rise to waves with the classical sech asymptotics in (2.35). However, when (cid:37) − /h (cid:28) ,cubic terms enter at leading order, and so one instead obtains an equation of Gardner or mKdV-KdVtype. An important consequence of this construction is that the O ( ε ) remainder terms in (2.35)and (2.36) are exponentially decaying and exhibit the same scaling of the spatial variable as theleading-order part. Note also that the regularity of the solutions is not stated by Nilsson, but followsfrom a standard bootstrapping argument.Theorem 2.11 fixes β but allows the dimensional parameters to vary. While convenient for provingexistence, this choice is not ideal for stability analysis: two waves on one of these curves may notnecessarily solve the same physical problem. The general theory in [26, 51] instead asks for a familyof bound states parameterized by c , with the remaining dimensional parameters held constant. Givena choice of parameters ( ρ ±∗ , d ±∗ , σ ∗ , c ∗ ) , we therefore let(2.37) ( β c , λ c ) := (cid:18) σ ∗ d + ∗ ρ −∗ c , − g (cid:74) ρ ∗ (cid:75) d + ∗ ρ −∗ c (cid:19) , ε A c := (cid:112) λ c − λ for | c − c ∗ | (cid:28) . The first of these parameterizes a segment of the straight line joining ( β ∗ , λ ∗ ) to the origin in the ( β, λ ) -plane, while the second expresses the bifurcation parameter ε from Theorem 2.11 in terms of c .The next two corollaries convert Theorem 2.11 to statements on bound states indexed by c . Inparticular, they prove that [51, Assumption 5] is satisfied. Corollary 2.13 (KdV bound states) . Let ( ρ ±∗ , d ±∗ , σ ∗ , c ∗ ) be given so that (cid:37) ∗ − /h ∗ (cid:54) = 0 and thecorresponding non-dimensional parameters ( β ∗ , λ ∗ ) lies in Region A. There exists an open interval I (cid:51) c ∗ and a family of bound states { U A c } c ∈ I ⊂ O ∩ W having the non-dimensional parametervalues ( β c , λ c ) given by (2.37) . The free surface profile is η A c := η A ε A c ; β c for c ∈ I . Moreover, { U A c } satisfies [51, Assumption 5] in that the following holds. (a) The mapping c ∈ I (cid:55)→ U A c ∈ O ∩ W is C . (b) For all c ∈ I , U A c ∈ Dom T (cid:48) (0) ∩ Dom
J IT (cid:48) (0) , U A c , J IT (cid:48) (0) U c ∈ Dom T (cid:48) (0) | W . (c) Each U A c is nontrivial in that T (cid:48) (0) U c (cid:54)≡ for c ∈ I . (d) The waves are localized in that lim inf | s |→∞ (cid:107) T ( s ) U A c − U A c (cid:107) X > . Proof.
Let ( ρ ±∗ , d ±∗ , σ ∗ , c ∗ ) be given as above and assume that the corresponding ( β ∗ , λ ∗ ) satisfy β ∗ > β and < λ ∗ − λ (cid:28) . Then for all | c − c ∗ | (cid:28) , the dimensional parameters meet thehypotheses of Theorem 2.11(a), and so may simply take U A c := u ε A c ; β c for β c and ε A c defined accordingto (2.37).The free surface profile from (2.35) is constructed as a solution to a second-order ODE that isa homoclinic to . It can be verified directly that the origin is a saddle point, and hence η A ε ; β isexponentially localized, with uniform decay rate on compact subsets of parameter space. Moreover,due to the translation invariance, the profile is of class C ∞ . In particular, it is clearly an elementof X k for all k ≥ / . Solving the kinematic condition, we see that the corresponding ψ = ψ A ε ; β is likewise smooth and an element of X k for all k ≥ / . Part (a) now follows from the smoothdependence of u ε ; β on ( ε, β ) . Part (b) certainly holds in view of the (arbitrarily high) regularity ofthe bound states. Finally, parts (c) and (d) are obvious given the form of η A c . (cid:3) An identical argument applied to the family of waves in Theorem 2.11(b) yields the following.
Corollary 2.14 (Gardner bound states) . Let ( ρ ±∗ , d ±∗ , σ ∗ , c ∗ ) be given so that the corresponding ( β ∗ , λ ∗ ) lies in Region A and | (cid:37) ∗ − /h ∗ | (cid:104) | λ ∗ − λ | / . There exists an open interval I (cid:51) c ∗ and twofamilies of bound states { U A ± c } c ∈ I ⊂ O ∩ W having the non-dimensional parameter values ( β A c , λ A c ) given by (2.37) and with the remaining parameters fixed. They satisfy [51, Assumption 5] and thecorresponding free surface is given by η A ± c := η A ε A c ; β c ,κ A c , ± for κ A c := 1 ε A c (cid:18) (cid:37) ∗ − h ∗ (cid:19) , c ∈ I . Consider now the situation where ( β, λ ) is contained in Region C, which is a neighborhood of thecurve Γ . Nilsson uses a center manifold reduction method to construct traveling waves, this timebifurcating from the point ( β , λ ) . Setting γ := ( (cid:37) + h ) / , one can show using the parameterizationof Γ that for all δ ∈ R , the point(2.38) β = β + 2(1 + δ ) γε , λ = λ + γε is contained in Region C for all < ε (cid:28) . When δ > , it lies below Γ and for δ < , it lies above.At ε = 0 , this gives the critical parameter value ( β , λ ) where we recall that is an eigenvalue ofmultiplicity . The resulting reduced equation on the center manifold thus has a four-dimensionalphase space. When (cid:37) − /h = O (1) as ε (cid:38) , after performing a rescaling and truncation, weobtain the ODE(2.39) Z (cid:48)(cid:48)(cid:48)(cid:48) − δ ) Z (cid:48)(cid:48) + Z − γ − / (cid:18) (cid:37) − h (cid:19) Z = 0 . This equation arises in the study of capillary-gravity waves beneath vacuum in the critical surfacetension regime as well as a modeling the buckling of elastic struts [3]. Analysis in [18, 15] showsthat, at δ = 0 , there is a primary homoclinic solution Z to (2.39) that is unimodal, even, andexponentially localized. Moreover, there is a smooth one-parameter family of homoclinic orbits { Z δ } δ defined for δ ≥ and − (cid:28) δ < that bifurcates from Z . These solutions are transverselyconstructed , in that the stable and unstable manifolds of the zero equilibrium of (1.9) intersecttransversely at Z = Z δ (0) at the zero level set of the Hamiltonian energy. For δ ≥ , we have that Z δ is the unique (up to translation) homoclinic solution to (1.9) that is positive, even and monotonefor x > (see [3]). When − (cid:28) δ < , uniqueness is not known and Z δ has exponentially decayingoscillatory tails. In addition to the primary homoclinic orbits, there exists a “plethora” of othersolutions to (2.39) that take the form of multisolitons; see [15, 22]. Because these are multimodal,they are unlikely to be amenable to analysis through the general theory in [51] and so we will notconsider them here. NTERNAL WAVE STABILITY 17
On the other hand, if (cid:37) − /h = κε , for some κ (cid:54) = 0 , then upon rescaling and truncating toleading order, the reduced equation on the center manifold takes the form(2.40) Z (cid:48)(cid:48)(cid:48)(cid:48) − δ ) Z (cid:48)(cid:48) + Z − γ − / κZ − γ − (cid:18) (cid:37) + 1 h + 2( (cid:37) − γ (cid:19) Z = 0 . In [46, Appendix B], it is shown that, at δ = 0 , this ODE has both a positive and negative primaryhomoclinic solution, which we denote by Z κ, ± . As in the non-resonant case, these are exponentiallylocalized, unique up to translation (for the fixed sign), and because they are transversely constructed,they persists for | δ | (cid:28) . Let the corresponding families be denoted { Z δ ; κ, ± } .We now state Nilsson’s results for this case reformulated in the style of Corollaries 2.13 and 2.14.Let ( ρ ±∗ , d ±∗ , σ ∗ , c ∗ ) be given so that the corresponding ( β ∗ , λ ∗ ) lies in Region C. In view of (2.38),we define(2.41) ε C c := (cid:18) λ c − λ γ ∗ (cid:19) / , δ c := β c − β γ ∗ ( ε C c ) − for | c − c ∗ | (cid:28) , with ( β c , λ c ) given in (2.37). The existence of bound states is then summarized in the followinglemma. Lemma 2.15 (Region C bound state) . Let ( ρ ±∗ , d ±∗ , σ ∗ , c ∗ ) be given so that (cid:37) ∗ − /h ∗ (cid:54) = 0 and thecorresponding non-dimensional parameters ( β ∗ , λ ∗ ) lie in Region C with < ε C c ∗ (cid:28) . (a) There exists an open interval I (cid:51) c ∗ and a family of bound states { U C c } c ∈ I ⊂ O ∩ W having the non-dimensional parameter values ( β c , λ c ) and satisfying [51, Assumption 5] . Thecorresponding free surface profile takes the form (2.42) η C c = ε d + √ γZ δ (cid:18) ε · d + (cid:19) + O (cid:0) ε (cid:1) in X k with ε = ε C c and δ = δ C c given by (2.41) , d + = d + ∗ , and γ = γ ∗ . (b) Suppose that | (cid:37) ∗ − /h ∗ | (cid:104) ( (cid:15) C c ∗ ) (cid:28) . Then there exists an open interval I (cid:51) c ∗ and twofamilies of bound states { U C ± c } c ∈ I ⊂ O ∩ W having the non-dimensional parameter values ( β c , λ c ) and satisfying [51, Assumption 5] . The corresponding free surface profile takes theform (2.43) η C ± c = ε d + √ γZ δ ; κ, ± (cid:18) ε · d + (cid:19) + O (cid:0) ε (cid:1) in X k for κ = κ C c := 1 ε c (cid:18) (cid:37) ∗ − h ∗ (cid:19) , with ε = ε C c and δ = δ C c given by (2.41) , d + = d + ∗ , and γ = γ ∗ .Remark . While we will carry out many of the calculations for both families { U C c } and { U C ± c } ,we only obtain a stability result for the former. In the latter case, we find that the rescaled linearizedaugmented potential does not converge precisely to the linearization of (2.39), which obstructs thespectral analysis in the next section; see Lemma 3.10(b).We conclude this section by noting that Nilsson also proves the existence of many types of travelingwaves with ( β, λ ) in a neighborhood of the bifurcation curve(2.44) Γ = { ( β ( iξ ) , λ ( iξ )) : ξ ∈ [0 , ∞ ) } , with ( β ( ξ ) , λ ( ξ )) given by (2.34). The stability of these solutions will be the subject of a forthcomingwork. Spectral analysis
Observe that if u ( t ) = T ( ct ) U is a traveling wave for the bound state U ∈ O ∩ W and wave speed c ∈ R , then necessarily by (2.25) and Lemma 2.9(f) we haved u d t = cT (cid:48) (0) U = J D E ( U ) . Combining this with (2.30), we obtain the steady equationD E ( U ) = c D P ( U ) , where we have used that J is injective. This motivates us to consider the augmented Hamiltonian ,which for a fixed c is the functional E c ∈ C ∞ ( O ∩ V ; R ) given by E c ( u ) := E ( u ) − cP ( u ) . The above calculation shows that bound states are critical points of E c . It also suggests that one canconstruct such solutions as constrained extrema of the energy on level sets of the momentum, withthe wave speed a Lagrange multiplier. In order to exploit this connection, we must first understandbetter the second derivative of E c .With that in mind, this section is devoted to the quite difficult task of computing the spectrum ofthe linearized augmented Hamiltonian at either a shear flow or small-amplitude internal capillary-gravity wave. Here we will follow the general approach of Mielke [45], which was also the basis forthe calculation in [51]. The strategy has two steps. First, via the kinematic condition ψ is eliminatedin favor of η . Making this substitution in the definition of E c gives the so-called augmented potential V aug c = V aug c ( η ) , which proves to be much more amenable to analysis. In particular, we show inSection 3.1 that its second variation at a critical point is characterized by a certain second-ordernonlocal differential operator Q c ( η ) . As one might predict, Q c (0) is a Fourier multiplier whosesymbol is related directly to the dispersion relation (1.3).This is enough to characterize the continuous spectrum of D V aug c ( η ) when η is sufficiently smallamplitude; see Lemma 3.7. Determining the discrete spectrum, however, requires considerably moreeffort. Following Mielke, the second step is to conjugate Q c ( η ) with a rescaling S ε informed by theasymptotics of η discussed in Section 2.5. Briefly put, the idea here is to show that linearization andscaling almost commute. It is well known that in the shallow water regime, the internal wave systemcan be modeled by nonlinear dispersive PDEs such as KdV or Gardner. We seek to prove thatimposing this scaling on the linearized operator Q c ( η ) via conjugation by S ε will, to leading order,coincide with the linearization of the corresponding model equation. After a delicate calculation, wedo indeed find that in the long-wave limit ε (cid:38) , the rescaled operator S − ε Q c ( η ) S ε converges (in anappropriate sense) to the linearized steady KdV or Gardner equation in the case of Region A, andto the linearization of (2.39) or (2.40) in the case of Region C. This is the subject of Section 3.2. InSection 3.3, we prove that the spectrum of D V aug c is qualitatively the same as that of this limitingrescaled operator.Lastly, in Section 3.4 we take the hard-won information about the spectrum of D V aug c andtranslate it back to that of D E c . For U c one of the family of bound states described in Section 2.5,we confirm that D E c ( U c ) extends to a self-adjoint operator on X that has Morse index . This isthe final hypothesis in the general theory [51].3.1. The augmented potential and its derivatives. If u ∗ = ( η ∗ , ψ ∗ ) is a critical point of E c ,then in particular D ψ E ( u ∗ ) = c D ψ P ( u ∗ ) . Because V is independent of ψ and A ( η ) is self-adjoint,we see that D ψ E ( u ) ˙ ψ = (cid:90) R ˙ ψA ( η ) ψ d x. Combining this with (2.31) we find that ψ ∗ can be uniquely determined from η ∗ via(3.1) ψ ∗ ( η ) := − cA ( η ) − η (cid:48) . NTERNAL WAVE STABILITY 19
Note that ψ ∗ also depends on c , but in this section the wave speed will be fixed, so there is no harmin suppressing it. In fact it will turn out to be easier to work with ϕ ∗ rather than ψ ∗ . So we recallfrom (2.5) and (2.4) that(3.2) ψ ∗ = ρ − ϕ ∗− − ρ + ϕ ∗ + , ϕ ∗± = ± cG ± ( η ) − η (cid:48) . When there is no risk of confusion, we will drop the ∗ subscripts to declutter the notation.Recall from Remark 2.4 that the coefficients a ± and a ± that arise in the first derivative formula(2.18) for G ± ( η ) can be alternatively be expressed as a ± ( η, φ ) := ∓ ( ∂ x H ± ( η ) φ ) | S , a ± := − ( ∂ y H ± ( η ) φ ) | S . Therefore, when they are evaluated at φ = ϕ ± , they give (up to a sign) the trace of the velocity fieldon S . Following [51, Section 6], we introduce the related functions(3.3) b ± := ∓ a ± ( η, ϕ ± ) − c, b ± := − a ± ( η, ϕ ± ) . This way, ( b ± , b ± ) represents the relative velocity in Ω ± restricted to the interface. Consequently, for η ∈ W , we have from (3.2) that b ± , b ± ∈ H ( R ) . Notice also that, because u represents a travelingwave, the kinematic condition (2.8) gives(3.4) b ± = η (cid:48) b ± . Differentiating (3.2), we find thatD ψ ∗ ( η ) ˙ η = ρ − D ϕ − ( η ) ˙ η − ρ + D ϕ + ( η ) ˙ η, D ϕ ± ( η ) ˙ η = ±(cid:104) c D ( G ± ( η ) − ) ˙ η, η (cid:48) (cid:105) ± cG ± ( η ) − ˙ η (cid:48) . On the other hand, (cid:104) D ( G ± ( η ) − ) ˙ η, η (cid:48) (cid:105) = − G ± ( η ) − (cid:104) D G ± ( η ) ˙ η, G ± ( η ) − η (cid:48) (cid:105) , and so we may infer from Lemma 2.3 and (3.2) that G ± ( η ) (cid:104) D ( G ± ( η ) − ) ˙ η, cη (cid:48) (cid:105) = ( a ± ( η, ± ϕ ± ) ˙ η ) (cid:48) − G ± ( η )( a ± ( η, ± ϕ ± ) ˙ η )= ± ( a ± ( η, ϕ ± ) ˙ η ) (cid:48) ∓ G ± ( η )( a ± ( η, ϕ ± ) ˙ η ) . Thus, D ϕ ± ( η ) ˙ η = G ± ( η ) − ( a ± ( η, ϕ ± ) ˙ η ) (cid:48) − a ± ( η, ϕ ± ) ˙ η ± cG ± ( η ) − ˙ η (cid:48) = ∓ G ± ( η ) − ( b ± ˙ η ) (cid:48) + b ± ˙ η, and hence D ψ ∗ ( η ) ˙ η = (cid:88) ± ρ ± G ± ( η ) − ( b ± ˙ η ) (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) =: S ˙ η − (cid:88) ± ± ρ ± b ± ˙ η (cid:124) (cid:123)(cid:122) (cid:125) =: T ˙ η . (3.5)Now, let the augmented potential be the functional V aug c ∈ C ∞ ( O ∩ V ; R ) given by(3.6) V aug c ( η ) := E c ( η, ψ ∗ ( η )) = min ψ E c ( η, ψ ) . While it is not immediately obvious, for small-amplitude waves the spectrum of D E c can bedetermined from that of D V aug c . We therefore devote the remainder of this subsection to studyingthe second variation of V aug c . In particular, we will derive an analytically tractable quadratic formrepresentation defined in terms of physical quantities.An essential ingredient in all of these calculations is having access to concise formulas for thevariations of the many nonlocal operators. First, we need the following elementary second derivativeformula for the Dirichlet–Neumann operators G ± . Here we use notation similar to that in [45, 51]. Lemma 3.1 (Second derivative of G ± ) . For all u = ( η, ψ ) ∈ O ∩ V and ˙ η ∈ V , it holds that (cid:90) R ψ (cid:10) D G ± ( η )[ ˙ η, ˙ η ] , ψ (cid:11) d x = (cid:90) R (cid:0) a ± ( u ) ˙ η + 2 a ± ( u ) ˙ ηG ± ( η ) (cid:0) a ± ( u ) ˙ η (cid:1)(cid:1) d x, (3.7) where a ± ( u ) := − a ± ( u ) (cid:48) a ± ( u ) , (3.8) and a ± , a ± are given by (2.19) .Proof. This is a straightforward though quite tedious calculation. (cid:3)
Far more involved is the second derivative of A ( η ) , a formula for which is given in the next lemma.As the proof is rather long but not especially deep, we delay it to Appendix A. Lemma 3.2 (Second derivative of A ) . For all u = ( η, ψ ) ∈ O ∩ V and ˙ η ∈ V , it holds that (cid:90) R ψ (cid:10) D A ( η )[ ˙ η, ˙ η ] , ψ (cid:11) d x = (cid:90) R (cid:16) a ( u ) ˙ η + 2 (cid:88) ± ρ ± a ± ( η, θ ± ) G ± ( η ) (cid:0) a ± ( η, θ ± ) ˙ η (cid:1) − M ( u ) ˙ η + 2 N ( u ) ˙ η (cid:17) ˙ η d x, (3.9) where we define the functions (3.10) θ ± ( u ) := G ± ( η ) − A ( η ) ψ, a ( u ) := (cid:88) ± ρ ± a ± ( η, θ ± ) , and linear operators L ± ( u ) ˙ η := − G ± ( η ) − (cid:0) a ± ( η, θ ± ) ˙ η (cid:1) (cid:48) + a ± ( η, θ ± ) ˙ η, L ( u ) := (cid:88) ± ρ ± L ± ( u ) (3.11) M ( u ) ˙ η := (cid:88) ± ρ ± (cid:0) a ± ( η, θ ± )( L ± ( u ) ˙ η ) (cid:48) + a ± ( η, θ ± ) G ± ( η ) L ± ( u ) ˙ η (cid:1) (3.12) N ( u ) ˙ η := (cid:88) ± ρ ± (cid:16) a ± ( η, θ ± ) (cid:0) A ( η ) G ± ( η ) − L ( u ) ˙ η (cid:1) (cid:48) + a ± ( η, θ ± ) A ( η ) L ( u ) ˙ η (cid:17) . (3.13) Remark . Formally setting ρ + = 0 and ρ − = 1 recovers the standard one-fluid model withnormalized density. We can see from (2.21) that this would imply A ( η ) = G − ( η ) , and so (3.9) mustagree with the second variation formula (3.7). Indeed, one can verify directly that θ − = ψ , so that L − ( u ) = − G − ( η ) − ∂ x a − ( u ) + a − ( u ) , a ( u ) = a − ( u ) , and hence N ( u ) = a − ( u ) ∂ x L − ( u ) + a − ( u ) G − ( η ) L − ( u ) = M ( u ) , giving back the one-fluid formula in [45, Proposition 2.1]. Lemma 3.4 (Second derivative of V aug c ) . For all ( η, ψ ∗ ( η )) ∈ O ∩ V and ˙ η ∈ V , it holds that D V aug c ( η )[ ˙ η, ˙ η ] = D η E c ( η, ψ ∗ ( η ))[ ˙ η, ˙ η ] − (cid:90) R ( S − T ) ˙ ηA ( η )( S − T ) ˙ η d x (3.14) where S and T are defined in (3.5) .Proof. Starting from the definition of V aug c in (3.6), we see thatD V aug c ( η ) ˙ η = D η E c ( u ∗ ) ˙ η + D ψ E c ( u ∗ ) D ψ ∗ ( η ) ˙ η = D η E c ( u ∗ ) ˙ η, NTERNAL WAVE STABILITY 21 where u ∗ = u ∗ ( η ) := ( η, ψ ∗ ( η )) . Note that the last equality follows from the fact that u ∗ is a criticalpoint of E c for all η . Differentiating again in η givesD V aug c ( η )[ ˙ η, ˙ η ] = D η E c ( u ∗ )[ ˙ η, ˙ η ] + D ψ D η E c ( u ∗ )[ Dψ ∗ ( η ) ˙ η, ˙ η ]= D η E c ( u ∗ )[ ˙ η, ˙ η ] − D ψ E c ( u ∗ )[ D ψ ∗ ( η ) ˙ η, D ψ ∗ ( η ) ˙ η ] . The potential energy is independent of ψ and the momentum is linear in ψ . Thus,D ψ E c ( u ∗ )[ D ψ ∗ ( η ) ˙ η, D ψ ∗ ( η ) ˙ η ] = D ψ K ( u ∗ )[ D ψ ∗ ( η ) ˙ η, D ψ ∗ ( η ) ˙ η ] = (cid:90) R D ψ ∗ ( η ) ˙ ηA ( η ) D ψ ∗ ( η ) ˙ η d x, which, from (3.5), implies (3.14). (cid:3) Lemma 3.5 (Quadratic form) . For all ( η, ψ ∗ ( η )) ∈ O ∩ V and c ∈ R , there is a self-adjoint linearoperator Q c ( η ) ∈ Lin( X ; X ∗ ) such that (3.15) D V aug c ( η )[ ˙ η, ˙ ζ ] = (cid:68) Q c ( η ) ˙ η, ˙ ζ (cid:69) X ∗ × X for all ˙ η, ˙ ζ ∈ V . It is given explicitly by Q c ( η ) ˙ η = − (cid:18) σ ˙ η (cid:48) (cid:104) η (cid:48) (cid:105) (cid:19) (cid:48) − (cid:16) g (cid:74) ρ (cid:75) + (cid:88) ± ± ρ ± b ± ( b ± ) (cid:48) (cid:17) ˙ η + (cid:88) ± ρ ± b ± (cid:0) G ± ( η ) − ( b ± ˙ η ) (cid:48) (cid:1) (cid:48) . (3.16) Remark . Taking ρ + = 0 and ρ − = 1 recovers the one-fluid problem, and it is straightforward tosee that formula (3.16) agrees with computation in [45, Theorem 3.3]. Proof.
We continue to write u ∗ := ( η, ψ ∗ ( η )) . Since the momentum is linear in η , we see thatD η E c ( u ∗ )[ ˙ η, ˙ η ] = D η K ( u ∗ )[ ˙ η, ˙ η ] + D η V ( u ∗ )[ ˙ η, ˙ η ]= 12 (cid:90) R ψ ∗ (cid:104) D A ( η )[ ˙ η, ˙ η ] , ψ ∗ (cid:105) d x − (cid:90) R g (cid:74) ρ (cid:75) ˙ η d x + (cid:90) R σ ( ˙ η (cid:48) ) (cid:104) η (cid:48) (cid:105) d x. (3.17)The latter two terms on the right-hand side above are already in the desired form. But, to understandthe first requires the formula for the second variation of A ( η ) derived in Lemma 3.2.In particular, notice that when θ ± defined in (3.10) is evaluated at u ∗ , it simplifies to θ ± ( u ∗ ) = − cG ± ( η ) − η (cid:48) = ∓ ϕ ± , and a ± ( η, θ ± ) = b ± + c , a ± ( η, θ ± ) = ± b ± . We further define S ± ( η ) ξ := G ± ( η ) − ( b ± ξ ) (cid:48) , T ± ( η ) ξ := ± b ± ξ, so that S ( η ) = (cid:80) ± ρ ± S ± ( η ) and T ( η ) = (cid:80) ± ρ ± T ± ( η ) . Making these substitution, we find from thesecond derivative formula (3.9) that (cid:90) R ψ ∗ (cid:104) D A ( η )[ ˙ η, ˙ η ] , ψ ∗ (cid:105) dx = (cid:88) ± ρ ± (cid:90) R (cid:0) ∓ ( b ± ) (cid:48) b ± ˙ η + T ± ˙ ηG ± ( η ) T ± ˙ η (cid:1) d x + (cid:90) R ( − ˙ η M ( u ∗ ) ˙ η + ˙ η N ( u ∗ ) ˙ η ) d x. Let us next look more closely at the two terms on the second line above. Observe first that L ± ( u ∗ ) ˙ η = − G ± ( η ) − (cid:0) ( b ± + c ) ˙ η (cid:1) (cid:48) ± b ± ˙ η = − (cid:0) T ± − S ± + cG ± ( η ) − ∂ x (cid:1) ˙ η L ( u ∗ ) = T − S − cA ( η ) − ∂ x , (3.18)where the second line follows from the first and (2.21). Because L ± and L will be evaluated at u ∗ throughout the calculation, we will suppress their arguments in the interests of readability. Using (3.18), we see that the operator M defined in (3.12) at the critical point satisfies (cid:90) R ˙ η M ˙ η d x = (cid:88) ± ρ ± (cid:90) R (cid:16) ( b ± + c )( L ± ˙ η ) (cid:48) ± b ± G ± ( η ) L ± ˙ η (cid:17) ˙ η d x = (cid:88) ± ρ ± (cid:90) R (cid:16) − (cid:0) ( b ± + c ) ˙ η (cid:1) (cid:48) ± G ± ( η ) b ± ˙ η (cid:17) L ± ˙ η d x = (cid:88) ± ρ ± (cid:90) R L ± ˙ ηG ± ( η ) L ± ˙ η d x, where again we are abbreviating M = M ( u ∗ ) . Substituting in the expression (3.18) and expandingyields (cid:90) R ˙ η M ˙ η d x = (cid:88) ± ρ ± (cid:90) R (cid:16) S ± ˙ ηG ± ( η ) S ± ˙ η − S ± ˙ ηG ± ( η ) T ± ˙ η + T ± ˙ ηG ± ( η ) T ± ˙ η (cid:17) d x + (cid:90) R (cid:16) c ˙ η (cid:48) A ( η ) − ˙ η (cid:48) + 2 c ˙ η (cid:48) ( S − T ) ˙ η (cid:17) d x. For later use, we compute (cid:90) R S ± ˙ ηG ± ( η ) T ± ˙ η d x = ± (cid:90) R G ± ( η ) − (cid:0) b ± ˙ η (cid:1) (cid:48) G ± ( η )( b ± ˙ η ) d x = ± (cid:90) R (cid:0) b ± ˙ η (cid:1) (cid:48) ( b ± ˙ η ) d x = ± (cid:90) R (cid:16) ( b ± ) (cid:48) b ± − b ± ( b ± ) (cid:48) (cid:17) ˙ η d x. Finally, in view of (3.13) and the formula for L in (3.18), we have that N = N ( u ∗ ) satisfies (cid:90) R ˙ η N ˙ η d x = (cid:88) ± ρ ± (cid:90) R (cid:16) ( b ± + c )( A ( η ) G ± ( η ) − L ˙ η ) (cid:48) ± b ± A ( η ) L ˙ η (cid:17) ˙ η d x = (cid:88) ± ρ ± (cid:90) R (cid:16) ( b ± + c )( A ( η ) G ± ( η ) − (cid:0) T − S − cA ( η ) − ∂ x ) ˙ η (cid:1) (cid:48) ± b ± A ( η )( T − S − cA ( η ) − ∂ x ) ˙ η (cid:17) ˙ η d x. Recalling that A ( η ) and G ± ( η ) − commute, continuing to simplify the right-hand side we obtain (cid:90) R ˙ η N ˙ η d x = − (cid:88) ± ρ ± (cid:90) R G ± ( η ) − (cid:0) ( b ± + c ) ˙ η (cid:1) (cid:48) A ( η )( T − S − cA ( η ) − ∂ x ) ˙ η d x + (cid:90) R T ˙ ηA ( η )( T − S − cA ( η ) − ∂ x ) ˙ η d x = (cid:90) R ( T − S − cA ( η ) − ∂ x ) ˙ ηA ( η )( T − S − cA ( η ) − ∂ x ) ˙ η d x = (cid:90) R (cid:16) D ψ ∗ ( η ) ˙ ηA ( η ) D ψ ∗ ( η ) ˙ η + 2 c ˙ η (cid:48) ( S − T ) ˙ η + c ˙ η (cid:48) A − ˙ η (cid:48) (cid:17) d x. NTERNAL WAVE STABILITY 23
Putting the above together and using Lemma 3.4, (3.17) and Lemma 3.2 we obtain D V aug c ( η )[ ˙ η, ˙ η ] = D η E c ( u ∗ )[ ˙ η, ˙ η ] − (cid:90) R D ψ ∗ ( η ) ˙ ηA ( η ) D ψ ∗ ( η ) ˙ η d x = (cid:90) R (cid:18) ψ ∗ (cid:104) D A ( η )[ ˙ η, ˙ η ] , ψ ∗ (cid:105) − g (cid:74) ρ (cid:75) ˙ η + σ ( ˙ η (cid:48) ) (cid:104) η (cid:48) (cid:105) (cid:19) d x − (cid:90) R D ψ ∗ ( η ) ˙ ηA ( η ) D ψ ∗ ( η ) ˙ η d x = (cid:90) R (cid:32) σ ( ˙ η (cid:48) ) (cid:104) η (cid:48) (cid:105) − (cid:16) g (cid:74) ρ (cid:75) + (cid:88) ± ± ρ ± b ± ( b ± ) (cid:48) (cid:17) ˙ η − (cid:88) ± ρ ± S ± ˙ ηG ± ( η ) S ± ˙ η (cid:33) d x, which leads to the formula Q c ( η ) claimed in (3.16). (cid:3) Following [45, Theorem 3.5], we can determine the continuous spectrum of Q c ( η ) as follows. Lemma 3.7 (Continuous spectrum) . Let u = ( η, ψ ) ∈ O ∩ V be given. Then the operator Q c ( η ) defined in (3.16) is self-adjoint on L ( R ) with domain H ( R ) . The continuous spectrum of Q c ( η ) isthe same as the one of Q c (0) , which is [ ν ∗ , + ∞ ) , where (3.19) ν ∗ := − g (cid:74) ρ (cid:75) (cid:18) − λ λ (cid:19) , for β ≥ β , − g (cid:74) ρ (cid:75) (cid:34) − λ max ξ ∈ R (cid:32)(cid:88) ± ρ ± ρ − d + ξ coth ( d ± ξ ) − βd ξ (cid:33)(cid:35) , for β < β . Proof.
The domain and the self-adjointness of Q c ( η ) follows from the regularity of η . The continuousspectrum of Q c ( η ) coincides with that of Q c (0) because η ( x ) → as | x | → ∞ . A direct computationyields that the Fourier symbol of Q c (0) is given by q c ( ξ ) := − g (cid:74) ρ (cid:75) (cid:34) − λ (cid:32)(cid:88) ± ρ ± ρ − d + ξ coth ( d ± ξ ) − βd ξ (cid:33)(cid:35) , which leads to the conclusion of the lemma. (cid:3) Remark . Observe that the symbol q c above recovers the dispersion relation in that d + ξ is a rootof (1.3) if and only if q c ( ξ ) = 0 .3.2. Rescaled operator.
We now execute the second step in the plan outlined at the start of thesection, namely using a long-wave rescaling to discern the leading-order form of the operator Q c ( η ) in the small-amplitude limit along the families of waves discussed in Section 2.5. Because we wishto exploit the fact that ( β, λ ) is close to the curve Γ or Γ , it is more convenient to perform thesecalculations working with the parameterization in [46]. With that in mind, let { Π ε } be a smoothcurve in the dimensional parameter space. For Region A, we assume that the corresponding β > β is fixed and λ = λ + ε , whereas for Region C, ( β, λ ) are given by (2.38) with δ fixed. To avoidcluttered notation, the dependence of ( ρ ± , d ± , σ, c ) on ε will be suppressed when there is no riskof confusion. Recall that the corresponding curves of traveling waves are denoted C A β , C A β,κ, ± , C C β,δ ,and C C β,δ,κ, ± .The main character in this analysis is the scaling operator S ε f := f (cid:18) ε · d + (cid:19) . Clearly S ε is a bounded isomorphism on H k ( R ) for all k ≥ with (cid:107) S ε (cid:107) Lin( H k ) = O ( ε − ) . Note that ∂ x and S ε satisfy the following commutation identities. ∂ x S ε = εd + S ε ∂ x , ∂ x S − ε = d + ε S − ε ∂ x , In particular, this shows that ∂ x S ε and ∂ x S − ε are uniformly bounded in Lin( H k +1 , H k ) for any k .From the existence theory in Section 2.5, the traveling wave profiles can be written(3.20) η ε =: ε m d + S ε ( (cid:101) η + (cid:101) r ε ) , (cid:101) r ε = O ( ε ) in W as ε (cid:38) , with m := for C A β and C C β,δ,κ, ± , for C A β,κ, ± , for C C β,δ .Note that in (3.20) we are continuing the practice of omitting superscripts and subscripts whenthey can be inferred from context. Thus, from (2.35) and (2.36) it follows that in Region A, ˜ η is ascaled KdV or Gardner soliton, while in Region C it is given by Z δ or Z δ,κ, ± . From the commutationidentities, we then have that η (cid:48) ε = ε m +1 S ε ( (cid:101) η (cid:48) + (cid:101) r (cid:48) ε ) . Abusing notation somewhat, let Q ε be the operator resulting from evaluating Q c at the parametervalues Π ε :(3.21) Q ε ( η ε ) := − ∂ x (cid:18) σ (cid:104) η (cid:48) ε (cid:105) ∂ x (cid:19) − (cid:16) g (cid:74) ρ (cid:75) + (cid:88) ± ± ρ ± b ± ε ( b ± ε ) (cid:48) (cid:17) + (cid:88) ± ρ ± b ± ε ∂ x G ± ( η ε ) − ∂ x b ± ε where b ± iε = b ± i ( η ε ) is a multiplication operator and η ε is from one of the families C A β , C A β,κ, ± , C C β,δ , or C C β,δ,κ, ± . Note that again the dependence of many quantities on ε is being suppressed. Our interestis the rescaled operator:(3.22) (cid:101) Q ε ( η ε ) := 1 ε n d + c ρ − S − ε Q ε ( η ε ) S ε , where n = 2 in Region A and n = 4 in Region C. Conjugating by S ε imposes a long-wave scalingthat will, in the limit ε (cid:38) , converge to the linearized operator for the corresponding dispersivemodel equation. We are also non-dimensionalizing the problem in order to simplify the resultingexpressions. Lemma 3.9 (Expansion of (cid:101) Q ε ) . The operator (cid:101) Q ε defined in (3.22) admits the expansion (cid:101) Q ε ( η ε ) = (cid:101) Q ε (0) + (cid:101) R ε , where in Region A (3.23) (cid:101) R ε = − (cid:18) (cid:37) − h (cid:19) (cid:101) η + O ( ε ) for C A β − κ (cid:101) η − (cid:18) (cid:37) + 1 h (cid:19) (cid:101) η + O ( ε ) for C A β,κ, ± , in Lin( H k +2 , H k ) , and in Region C (3.24) (cid:101) R ε = − (cid:18) (cid:37) − h (cid:19) (cid:101) η + O ( ε ) for C C β,δ − κ (cid:101) η − (cid:18) (cid:37) + 1 h (cid:19) (cid:101) η + (1 − (cid:37) ) (cid:0) ∂ x ( (cid:101) η∂ x ) + (cid:101) η (cid:48)(cid:48) (cid:1) + O ( ε ) for C C β,δ,κ, ± , in Lin( H k +2 , H k ) . NTERNAL WAVE STABILITY 25
Proof.
Looking at its definition in (3.21), we see that Q ε ( η ε ) is the sum of a second-order differentialoperator (call it the surface tension term), a multiplication operator (the potential term), and afirst-order nonlocal operator (the nonlocal term). Rescaling the surface tension term yields(3.25) − ε n d + c ρ − S − ε ∂ x (cid:18) σ (cid:104) η (cid:48) ε (cid:105) ∂ x (cid:19) S ε = − ε − n ∂ x (cid:18) β (cid:104) ε m +1 ( (cid:101) η (cid:48) + (cid:101) r (cid:48) ε ) (cid:105) ∂ x (cid:19) . To understand the contribution of the potential term to (cid:101) Q ε ( η ε ) , we first denote the non-dimensionalizedand rescaled relative velocity field(3.26) b ± ε =: cS ε (cid:101) b ± , b ± ε =: cS ε (cid:101) b ± . From the kinematic boundary condition (3.4) we then have that (cid:101) b ± = ε k +1 (cid:101) η (cid:48) (cid:101) b ± . Hence − ε n d + c ρ − S − ε (cid:32) g (cid:74) ρ (cid:75) + (cid:88) ± ± ρ ± b ± ε ( b ± ε ) (cid:48) (cid:33) S ε = λε n − ε m − n +2 (cid:88) ± ± ρ ± ρ − (cid:101) b ± ( (cid:101) η (cid:48) (cid:101) b ± ) (cid:48) . The rescaling of the nonlocal term in Q ε ( η ε ) will require the most effort to expand. Towards thatend, we define the operator (cid:102) M ± ε ( η ε ) ∈ Lin( H k +2 , H k +1 ) by(3.27) (cid:102) M ± ε ( η ε ) := d + ε n S − ε ∂ x G ± ( η ε ) − ∂ x S ε . In particular, this means that(3.28) (cid:101) Q ε (0) = 1 ε n (cid:32) − ε β∂ x + λ + (cid:88) ± ρ ± ρ − ε n (cid:102) M ± ε (0) (cid:33) . Now, using the above calculations, we will analyze the difference operator (cid:101) R ε := (cid:101) Q ε ( η ε ) − (cid:101) Q ε (0)= − βε − n ∂ x (cid:20)(cid:18) (cid:104) ε m +1 ( (cid:101) η (cid:48) + (cid:101) r (cid:48) ε ) (cid:105) − (cid:19) ∂ x (cid:21) − ε m − n +2 (cid:88) ± ± ρ ± ρ − (cid:101) b ± (˜ η (cid:48) (cid:101) b ± ) (cid:48) + (cid:88) ± ρ ± ρ − (cid:16)(cid:101) b ± (cid:102) M ± ε ( η ε ) (cid:101) b ± − (cid:102) M ± ε (0) (cid:17) . (3.29)In view of (3.20) and (3.25), the first term on the right-hand side above is higher order:(3.30) − βε − n ∂ x (cid:20)(cid:18) (cid:104) ε m +1 ( (cid:101) η (cid:48) + (cid:101) r (cid:48) ε ) (cid:105) − (cid:19) ∂ x (cid:21) = O ( ε m − n +4 ) in Lin( H k +2 , H k ) . Consider the remaining two terms in (3.29). Notice that for any f ∈ H k +2 we have F (cid:16) (cid:102) M ± ε (0) f (cid:17) ( ξ ) = d + ε n εd + F (cid:0) ∂ x G ± (0) − ∂ x S ε f (cid:1) (cid:18) εd + ξ (cid:19) = d + ε n m ± (cid:18) εd + ξ (cid:19) (cid:98) f ( ξ ) where m ± ( ξ ) := − ξ coth( d ± ξ ) is the symbol for ∂ x G ± (0) − ∂ x . Thus (cid:102) M ± ε (0) is indeed a Fouriermultiplier and its symbol is given by(3.31) (cid:101) m ± ε ( ξ ) := − ε n εξ tanh( d ± εξ/d + ) . As an immediate consequence, it follows that (cid:13)(cid:13)(cid:13)(cid:13) ε n (cid:102) M ± ε (0) + d + d ± (cid:13)(cid:13)(cid:13)(cid:13) Lin( H k +2 ,H k ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) (cid:104) · (cid:105) (cid:18) ε n (cid:101) m ± ε + d + d ± (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ (cid:46) ε . (3.32)In other words, ε n (cid:102) M ± ε (0) is to leading order the multiplication operator − d + /d ± in Lin( H k +2 , H k ) . To estimate the scaled relative velocity, we observe that by (3.2)–(3.3) and (3.26), it holds that (cid:101) b ± = 1 c S − ε ( ∂ x Φ ε ± | S − c ) , where, as usual, Φ ε ± denotes the velocity potential. But expanding the Dirichlet–Neumann operator,we find that ϕ (cid:48)± = ± c∂ x (cid:0) G ± ( η ε ) − ∂ x η ε (cid:1) = ± c∂ x (cid:2) G ± (0) − η (cid:48) ε + (cid:10) D G ± (0) − η ε , η (cid:48) ε (cid:11)(cid:3) + O ( ε m ) in H k , ( ∂ x Φ ε ± ) | S = 11 + ( η (cid:48) ε ) (cid:0) ϕ (cid:48)± ± η (cid:48) ε G ± ( η ε ) ϕ ± (cid:1) = 11 + ( η (cid:48) ε ) (cid:0) ϕ (cid:48)± ± ( η (cid:48) ε ) (cid:1) = ± c∂ x (cid:2) G ± (0) − η (cid:48) ε + (cid:10) D G ± (0) − η ε , η (cid:48) ε (cid:11)(cid:3) + O ( ε m ) in H k . We can compute D G ± (0) − as (cid:10) D G ± (0) − η ε , f (cid:11) = − G ± (0) − (cid:10) D G ± (0) η ε , G ± (0) − f (cid:11) , and from Lemma 2.3, we see that (cid:10) D G ± (0) η ε , G ± (0) − ∂ x S ε f (cid:11) = ± ∂ x S ε (cid:2)(cid:0) S − ε ∂ x G ± (0) − ∂ x S ε f (cid:1) d + ε m (cid:101) η (cid:3) ± G ± (0) S ε (cid:18) εd + ( ∂ x f ) d + ε m (cid:101) η (cid:19) = ± ∂ x S ε ε m + n (cid:16) (cid:102) M ± ε (0) f (cid:17) (cid:101) η ± ε m +1 G ± (0) S ε ( (cid:101) η∂ x f ) . (3.33)Therefore (cid:101) b ± = S − ε (cid:2) ± ∂ x (cid:0) G ± (0) − η (cid:48) ε + (cid:10) D G ± (0) − η ε , η (cid:48) ε (cid:11)(cid:1) − (cid:3) + O ( ε m )= ± ε m d + (cid:2) S − ε ∂ x G ± (0) − ∂ x ( S ε (cid:101) η ) − S − ε ∂ x G ± (0) − (cid:10) D G ± (0) η ε , G ± (0) − ∂ x S ε (cid:101) η (cid:11)(cid:3) − O ( ε m )= ± ε m + n (cid:102) M ± ε (0) (cid:101) η − O ( ε m ) ∓ ε m d + S − ε ∂ x G ± (0) − (cid:104) ± ε m + n ∂ x S ε (cid:16) (cid:102) M ± ε (0) (cid:101) η (cid:17) (cid:101) η ± ε m +1 G ± (0) S ε ( (cid:101) η (cid:101) η (cid:48) ) (cid:105) = ± ε m + n (cid:102) M ± ε (0) (cid:101) η − − ε m +2 n (cid:102) M ± ε (0) (cid:16)(cid:16) (cid:102) M ± ε (0) (cid:101) η (cid:17) (cid:101) η (cid:17) − ε m +2 ∂ x ( (cid:101) η (cid:101) η (cid:48) ) + O ( ε m )= − ∓ ε m d + d ± (cid:101) η − ε m d d ± (cid:101) η + O ( ε m +2 ) in H k . (3.34)Hence for the second term on the right-hand side of (3.29) we have − ε m − n +2 (cid:88) ± ± ρ ± ρ − (cid:101) b ± ( (cid:101) η (cid:48) (cid:101) b ± ) (cid:48) = ε m − n +2 (1 − (cid:37) ) (cid:101) η (cid:48)(cid:48) + ε m − n +2 (cid:18) (cid:37) + 1 h (cid:19) (cid:2) (cid:101) η (cid:101) η (cid:48)(cid:48) + ( (cid:101) η (cid:48) ) (cid:3) + O ( ε m − n +2 ) in Lin( H k +2 , H k ) . (3.35)Using the expansion (3.34) for ˜ b ± also furnishes the estimate (cid:101) b ± (cid:102) M ± ε ( η ε ) (cid:101) b ± = (cid:102) M ± ε ( η ε ) ± ε m d + d ± (cid:104)(cid:101) η (cid:102) M ± ε ( η ε ) + (cid:102) M ± ε ( η ε ) (cid:101) η (cid:105) + ε m d d ± (cid:104)(cid:101) η (cid:102) M ± ε ( η ε ) (cid:101) η + (cid:101) η (cid:102) M ± ε ( η ε ) + (cid:102) M ± ε ( η ε ) (cid:101) η (cid:105) + O ( ε m − n ) (3.36) NTERNAL WAVE STABILITY 27 in Lin( H k +2 , H k ) . On the other hand, from the definition of (cid:102) M ± ε in (3.27) it follows that for all f ∈ H k +2 with (cid:107) f (cid:107) H k +2 = 1 , (cid:16) (cid:102) M ± ε ( η ε ) − (cid:102) M ± ε (0) (cid:17) f = d + ε n S − ε ∂ x (cid:0) G ± ( η ε ) − − G ± (0) − (cid:1) ∂ x S ε f = d + ε n S − ε ∂ x (cid:10) D G ± (0) − η ε , ∂ x S ε f (cid:11) + d + ε n S − ε ∂ x (cid:10) D G ± (0) − [ η ε , η ε ] , ∂ x S ε f (cid:11) + O ( ε m − n ) in H k . (3.37)Explicit calculation yields (cid:10) D G ± (0) − [ η ε , η ε ] , f (cid:11) = − G ± (0) − (cid:10) D G ± (0)[ η ε , η ε ] , G ± (0) − f (cid:11) + 2 G ± (0) − (cid:10) D G ± (0) η ε , G ± (0) − (cid:10) D G ± (0) η ε , G ± (0) − f (cid:11)(cid:11) . From (3.33) we have (cid:104) D G ± (0) η ε , G ± (0) − (cid:10) D G ± (0) η ε , G ± (0) − ∂ x S ε f (cid:11)(cid:11) = ∂ x S ε (cid:104)(cid:16) S − ε ∂ x G ± (0) − ∂ x S ε ε m + n (cid:16) (cid:102) M ± ε (0) f (cid:17) (cid:101) η (cid:17) d + ε m (cid:101) η (cid:105) + O ( ε m +1 )= ε m + n ) ∂ x S ε (cid:102) M ± ε (0) (cid:16)(cid:16) (cid:102) M ± ε (0) f (cid:17) (cid:101) η (cid:17) (cid:101) η + O ( ε m +1 ) in H k . Likewise, Lemma 3.1 allows us to estimate (cid:10) D G ± (0)[ η ε , η ε ] , G ± (0) − f (cid:11) = O ( ε m +1 ) in H k . Substituting the above into (3.37) yields (cid:16) (cid:102) M ± ε ( η ε ) − (cid:102) M ± ε (0) (cid:17) f = ∓ d + ε n S − ε ∂ x G ± (0) − ∂ x S ε (cid:16) ε m + n (cid:102) M ± ε (0) f (cid:17) (cid:101) η + d + ε n S − ε ∂ x G ± (0) − ∂ x S ε (cid:16) ε m + n ) (cid:102) M ± ε (0) (cid:16) (cid:102) M ± ε (0) f (cid:17) (cid:101) η (cid:17) (cid:101) η ∓ d + ε n S − ε ∂ x S ε (cid:0) ε m +1 (cid:101) η∂ x f (cid:1) + O ( ε m − n +1 )= ∓ ε m + n (cid:102) M ± ε (0) (cid:16)(cid:16) (cid:102) M ± ε (0) f (cid:17) (cid:101) η (cid:17) ∓ ε m − n +2 ∂ x ( (cid:101) η∂ x f )+ ε m + n ) (cid:102) M ± ε (0) (cid:16) (cid:102) M ± ε (0) (cid:16) (cid:102) M ± ε (0) f (cid:17) (cid:101) η (cid:17) (cid:101) η + O ( ε m − n +1 )= ∓ ε m − n d d ± (cid:101) ηf ∓ ε m − n +2 ∂ x ( (cid:101) η∂ x f ) − ε m − n d d ± (cid:101) η f + O ( ε m − n +1 ) , (3.38)in H k . Using this, the previous estimate (3.36) becomes (cid:101) b ± (cid:102) M ± ε ( η ε ) (cid:101) b ± f = (cid:102) M ± ε (0) f ± ε m d + d ± (cid:101) η (cid:102) M ± ε (0) f ± ε m d + d ± (cid:102) M ± ε (0) (cid:101) ηf + ε m d d ± (cid:104)(cid:101) η (cid:102) M ± ε (0) (cid:101) ηf + (cid:101) η (cid:102) M ± ε (0) f + (cid:102) M ± ε (0) (cid:101) η f (cid:105) ∓ ε m − n d d ± (cid:101) ηf ∓ ε m − n +2 ∂ x ( (cid:101) η∂ x f ) − ε m − n d d ± (cid:101) η f − ε m − n +2 d + d ± [ (cid:101) η∂ x ( (cid:101) η∂ x f ) + ∂ x ( (cid:101) η∂ x ( (cid:101) ηf ))] + O ( ε m − n +1 ) , in H k . We can simplify further by applying (3.32), which results in (cid:101) b ± (cid:102) M ± ε ( η ε ) (cid:101) b ± f = (cid:102) M ± ε (0) f ∓ ε m − n d d ± (cid:101) ηf − ε m − n d d ± (cid:101) η f ∓ ε m − n +2 ∂ x ( (cid:101) η∂ x f )+ O ( ε m − n +1 ) in H k . Therefore in computing the third term on the right-hand side of (3.29) we find (cid:88) ± ρ ± ρ − (cid:16)(cid:101) b ± (cid:102) M ± ε ( η ε ) (cid:101) b ± − (cid:102) M ± ε (0) (cid:17) f = 3 ε m − n (cid:88) ± ∓ ρ ± ρ − d d ± (cid:101) ηf − ε m − n (cid:88) ± ρ ± ρ − d d ± (cid:101) η f + ε m − n +2 (cid:88) ± ∓ ρ ± ρ − ∂ x ( (cid:101) η∂ x f ) + O ( ε m − n +1 )= − ε m − n (cid:18) (cid:37) − h (cid:19) (cid:101) ηf − ε m − n (cid:18) (cid:37) + 1 h (cid:19) (cid:101) η f + ε m − n +2 (1 − (cid:37) ) ∂ x ( (cid:101) η∂ x f ) + O ( ε m − n +1 ) , in H k . Taken together with (3.30) and (3.35), this yields the claimed expansion for (cid:101) R ε . (cid:3) Let us now look more closely at the leading-order part of (cid:101) Q ε ( η ε ) , which by the above lemma is theFourier multiplier (cid:101) Q ε (0) . Analyzing its symbol will allow us to infer that it has a point-wise limitas ε (cid:38) . Near the critical Bond number, however, there is a degeneracy that causes the limitingoperator to be fourth order. Combining this with the previous result, we obtain the following. Lemma 3.10 (Limiting rescaled operator) . Consider the rescaled operator (cid:101) Q ε ( η ε ) given by (3.22) . (a) Suppose that β > β and λ = λ + ε lies in Region A. Then for any k > / and ζ ∈ H k +2 , (cid:107) (cid:101) Q ε ( η ε ) ζ − (cid:101) Q ζ (cid:107) H k −→ as ε (cid:38) , where the operator ˜ Q ∈ Lin( H k +2 , H k ) is given by (cid:101) Q = − ( β − β ) ∂ x + 1 − (cid:18) (cid:37) − h (cid:19) (cid:101) η for C A β − ( β − β ) ∂ x + 1 − κ (cid:101) η − (cid:18) (cid:37) + 1 h (cid:19) (cid:101) η for C A β ; κ, ± . (b) Suppose that ( β, λ ) lie in Region C and are given by (2.38) for a fixed δ < . Then for any k > / and ζ ∈ H k +4 , (cid:107) (cid:101) Q ε ( η ε ) ζ − (cid:101) Q ζ (cid:107) H k −→ as ε (cid:38) , where the operator (cid:101) Q ∈ Lin( H k +4 , H k ) is given by (cid:101) Q = γ∂ x − δ ) γ∂ x + γ − (cid:18) (cid:37) − h (cid:19) (cid:101) η for C C β,δ γ∂ x − δ ) γ∂ x + γ − κ (cid:101) η − (cid:18) (cid:37) + 1 h (cid:19) (cid:101) η + (1 − (cid:37) ) (cid:0) ∂ x ( (cid:101) η∂ x ) + (cid:101) η (cid:48)(cid:48) (cid:1) for C C β ; κ,δ, ± . Proof.
Fix k > / . Recall that (cid:101) Q ε ( η ε ) = (cid:101) Q ε (0) + (cid:101) R ε , where (cid:101) Q ε (0) is given in (3.28). We havealready seen in Lemma 3.9 that (cid:101) R ε has a uniform limit in Lin( H k +2 , H k ) as ε (cid:38) . From (3.31), itis clear that (cid:101) Q ε (0) is a Fourier multiplier: for all f ∈ H k +2 , F (cid:16) (cid:101) Q ε (0) f (cid:17) ( ξ ) = 1 ε n (cid:32) ε βξ + λ − (cid:88) ± ρ ± ρ − εξ coth (cid:18) d ± d + εξ (cid:19)(cid:33) (cid:98) f ( ξ ) =: (cid:101) q ε ( ξ ) (cid:98) f ( ξ ) . NTERNAL WAVE STABILITY 29
Consider the point-wise limit of the symbol (cid:101) q ε as ε (cid:38) . Here it is important to keep in mind thatthe dimensional parameters are moving along the curve { Π ε } and λ (cid:38) λ in this limit. Therefore,we write (cid:101) q ε ( ξ ) = ε − n ( β − β ) ξ + λ − λ ε n + 1 ε n (cid:32) β ( εξ ) + λ − (cid:88) ± ρ ± ρ − εξ coth (cid:18) d ± d + εξ (cid:19)(cid:33) =: ε − n ( β − β ) ξ + λ − λ ε n + r ( εξ ) ε n . Taylor expanding r near (cid:101) ξ := εξ = 0 yields that r ( (cid:101) ξ ) = β (cid:101) ξ + λ − (cid:88) ± ρ ± ρ − (cid:101) ξ coth (cid:18) d ± d + (cid:101) ξ (cid:19) = γ (cid:101) ξ + O ( (cid:101) ξ ) as (cid:101) ξ → . (3.39)For Region A, we have n = 2 and λ = λ + ε , and hence for each fixed ξ ∈ R , (cid:101) q ε ( ξ ) −→ ( β − β ) ξ + 1 as ε (cid:38) . On the other hand, in Region C we have n = 4 with ( β, λ ) given by (2.38). Again, fixing ξ we thenhave that the limiting symbol is (cid:101) q ε ( ξ ) −→ γξ + 2(1 + δ ) γξ + γ as ε (cid:38) . Combining these expressions for the limiting symbol with the asymptotics of (cid:101) R ε from (3.23) and(3.24), the formulas for (cid:101) Q in (a) and (b) now follow. (cid:3) Spectrum of the linearized augmented potential.
Using the limiting behavior derivedabove, we will now characterize the spectrum of the Q ε ( η ε ) . It is worth reiterating that an essentialchallenge in this analysis is that the operator converges point-wise to Q (0) whose essential spectrumis [0 , ∞ ) . It is for this reason that we introduced the rescaled operator (cid:101) Q ε ( η ε ) , since by Lemma 3.10converges (again only point-wise) to (cid:101) Q , which has a gap between the positive essential spectrumand . Spectral analysis in Region A.
We start by deriving the spectral properties of Q ε ( η ε ) for the strongsurface tension waves with parameters ( β, λ ) in Region A. Lemma 3.11.
In the setting of Lemma 3.10 (a) , the limiting rescaled operator (cid:101) Q satisfies (3.40) ess spec (cid:101) Q = [1 , ∞ ) , spec (cid:101) Q = {− (cid:101) ν , } ∪ (cid:101) Λ where the first two eigenvalues − (cid:101) ν < and are both simple with corresponding eigenfuctions (cid:101) φ and (cid:101) φ = (cid:101) η (cid:48) , respectively; and there exists ν ∗ > such that (cid:101) Λ ⊂ [ ν ∗ , ∞ ) .Proof. This is a classical result on linear Schrödinger operators, and can be found, for example, in[6]. The fact that − (cid:101) ν and are all simple follows from the theory of ODEs: the Wronskian of two L solutions to the eigenvalue problem (cid:101) Q f = (cid:101) νf is necessarily . (cid:3) Using a similar argument as [45, Theorem 4.3], we then have the following result.
Theorem 3.12 (Spectrum in Region A) . Let the assumptions of Lemma 3.10 (a) hold. For each a ∈ (0 , ν ∗ ) there exists some ε > such that for all ε ∈ (0 , ε ) the operator Q ε ( η ε ) satisfies ess spec Q ε ( η ε ) ⊂ [ ε c ρ − /d + , ∞ ) , spec Q ε ( η ε ) = {− ν , } ∪ Λ , where Λ ⊂ [ aε c ρ − /d + , ∞ ) , and ν = ε c ρ − d + (cid:101) ν + o ( ε ) as ε (cid:38) . The first two eigenvalues ν := − ν < and ν := 0 are simple with the associated eigenfunctionstaking the form φ i = S ε (cid:101) φ i + o (1) in H k as ε (cid:38) .Proof. From Lemma 3.9 we see that it suffices to prove that the operator Q ε := (cid:101) Q ε (0) + (cid:101) R , with (cid:101) R defined by (3.23) with ε = 0 , has exactly two simple eigenvalues lying in ( −∞ , a ) thatconverge to (cid:101) ν i respectively for i = 1 , . It is clear that Q ε is self-adjoint. Note that Q ε may not have0 as an exact eigenvalue, but this does hold for (cid:101) Q ε ( η ε ) .Firstly, from Lemma 3.10 (a) it follows that(3.41) (cid:107) ( Q ε − (cid:101) ν i ) (cid:101) φ i (cid:107) H k ≤ Cε (cid:107) (cid:101) φ i (cid:107) H k . Therefore Q ε admits spectral values close to (cid:101) ν i with O ( ε ) distance.Now we consider a sequence { ( ν ε j , φ ε j ) } of eigenpairs of Q ε j with ν ε j ∈ ( −∞ , a ) and ε j (cid:38) as j → ∞ . Our goal is to prove the compactness of the eigenpair sequence and confirm that the limitmust be an eigenpair of (cid:101) Q .We normalize so that (cid:107) φ ε j (cid:107) H k = 1 . Note that (cid:107) (cid:101) η (cid:107) W N, ∞ ≤ C N for any N ≥ . Moreover fromthe proof of Lemma 3.9 we see that (cid:101) Q ε (0) − is positive semi-definite. From this we know thatthe spectrum of Q ε is bounded below: spec Q ε ⊂ [1 − C k , ∞ ) . Since (cid:101) η decays exponentially, wehave that ess spec Q ε = [1 , ∞ ) . Thus spec Q ε ∩ [1 − C k , a ] consists of discrete eigenvalues of finitemultiplicity. By definition,(3.42) (cid:16) (cid:101) Q ε j (0) − ν ε j (cid:17) φ ε j = − (cid:101) R φ ε j . Since ν ε j ∈ [1 − C k , a ] , from the proof of Lemma 3.9, the Fourier symbol of operator on the left-handside is (cid:101) q ε j ( ξ ) − ν ε j ≥ (cid:101) q ( ξ ) − a = ( β − β ) ξ + 1 − a ≥ δ ∗ (1 + ξ ) for some δ ∗ > independent of ε j . This uniform ellipticity property allows us via bootstrapping toobtain the bound (cid:107) φ ε j (cid:107) H k +4 ≤ C ∗ from some universal constant C ∗ > .To obtain compactness of the sequence { φ ε j } in H k +2 , we proceed to prove a uniform decayestimate. Given an exponential weight w := cosh( α · ) for some α > , we see that for any Schwartzfunction f , F (cid:20) w (cid:16) (cid:101) Q ε j (0) − ν ε j (cid:17) − f (cid:21) ( ξ ) = 12 (cid:34) (cid:98) f ( ξ + iα ) (cid:101) q ε j ( ξ + iα ) − ν ε j + (cid:98) f ( ξ − iα ) (cid:101) q ε j ( ξ − iα ) − ν ε j (cid:35) . Taking α < (1 − a ) / ( β − β ) it follows that sup | Im ξ |≤ α (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) q ε j ( ξ ± iα ) − ν ε j (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∗ , for some C ∗ > . Therefore (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) (cid:101) Q ε j (0) − ν ε j (cid:17) − (cid:13)(cid:13)(cid:13)(cid:13) Lin( L w ) ≤ C ∗ , where L w := { f ∈ L : wf ∈ L } is the weighted L space corresponding to w . Hence from (3.42), (cid:107) φ ε j (cid:107) L w ≤ C ∗ (cid:107) (cid:101) R φ ε j (cid:107) L w ≤ C ∗ (cid:107) w (cid:101) R (cid:107) L ∞ (cid:107) φ ε j (cid:107) L ≤ C ∗ (cid:107) w (cid:101) R (cid:107) L ∞ (cid:46) . Thus { φ ε j } is bounded in H k +4 ∩ L w , which is compactly embedded in H k +2 . Hence up to asubsequence, as j → ∞ , ν ε j → ν ∗ ∈ ( −∞ , a ] and φ ε j → φ ∗ in H k +2 with (cid:107) φ ∗ (cid:107) H k = 1 . Moreover, (cid:101) Q φ ∗ = ν ∗ φ ∗ , which indicates that φ ∗ = (cid:101) φ i for some i = 1 , .Finally we check the convergence of the corresponding spectral projections. Set P ε to be thespectral projection for Q ε associated with the interval [1 − C k , a ] . From (3.41), there exists ε > NTERNAL WAVE STABILITY 31 such that dim
Rng P ε ≥ for ε ∈ (0 , ε ) . Also P ε = (cid:80) N ε i =1 (cid:104) · , φ i,ε (cid:105) φ i,ε for some finite integer N ε and orthonormal eigenbasis { φ i,ε } N ε i =1 . Were there a sequence ε j (cid:38) such that N ε j ≥ , then itwould contradict the above convergence result. Therefore, for all ε sufficiently small, it must be that N ε = 2 . We can then conclude that φ i,ε → (cid:101) φ i in H k . (cid:3) Spectral analysis in Region C.
The same argument can also be applied to the near critical surfacetension waves with ( β, λ ) in Region C. On the solution curve C C β,δ , we have that (cid:101) η satisfies(3.43) γ∂ x (cid:101) η − δ ) γ∂ x (cid:101) η + γ (cid:101) η − (cid:18) (cid:37) − h (cid:19) (cid:101) η = 0 . Direct computation shows that the Green’s function of [ ∂ x − δ ) ∂ x + 1] − decays like e − s | x | as x → ±∞ , where(3.44) s := (cid:40) (cid:113) δ − (cid:112) δ (2 + δ ) , δ ≥ , (cid:112) | δ | / , δ < , indicating that (cid:101) η is exponentially localized. Therefore, invoking the Weyl theorem on continuousspectrum, we know that(3.45) ess spec (cid:101) Q = [ γ, ∞ ) when δ > − . Note that the operator (cid:101) Q is self-adjoint in L ( R ) with domain H k +4 ( R ) . Therefore, its spectrumis confined to the real line. Standard ODE theory shows that any eigenvalue of (cid:101) Q has geometricmultiplicity ≤ .By setting Z := γ (cid:0) (cid:37) − h (cid:1) (cid:101) η , equation (3.43) becomes (1.9) Z (cid:48)(cid:48)(cid:48)(cid:48) − δ ) Z (cid:48)(cid:48) + Z − Z = 0 , which leads us to study(3.46) Q δ := ∂ x − δ ) ∂ x + 1 − Z δ viewed as an unbounded operator on L ( R ) with domain H ( R ) . While more exotic than theSchrödinger operator encountered in Region A, the spectral properties of this Q δ for δ > − havebeen studied by Sandstede [47]. We quote an important results of his below. Lemma 3.13 (Sandstede [47]) . Let δ > − and Z δ be a homoclinic solution of (1.9) , and considerthe linearized operator Q δ given by (3.46) . (i) Q δ has at least one negative eigenvalue. (ii) If Z δ is transversely constructed, then zero is a simple eigenvalue of Q δ . Moreover, when δ isvaried, the number of negative eigenvalues remains constant until Z δ ceases to be transverselyconstructed. (iii) In particular, for δ ≥ or − (cid:28) δ < and consider Z δ being a transversely constructedprimary homoclinic orbit. Then Q δ has exactly one negative eigenvalue. That is, the spectrumof Q δ takes the form ess spec Q δ = [1 , ∞ ) , spec Q δ = {− (cid:101) ν , } ∪ (cid:101) Λ where − (cid:101) ν < and are both simple with corresponding eigenfuctions (cid:101) φ and (cid:101) φ = Z (cid:48) δ ,respectively; and there exists ν ∗ > such that (cid:101) Λ ⊂ [ ν ∗ , ∞ ) . With these provisions, we obtain the following theorem of the spectrum of the augmented potentialin Region C. The proof is very similar to the one for Theorem 3.12, and hence we omit it.
Theorem 3.14 (Spectrum in Region C) . Let the assumptions of Lemma 3.10 (b) hold. Let (cid:101) ν, ν ∗ and (cid:101) φ , given as in Lemma 3.13 (iii) . Then for each a ∈ (0 , ν ∗ ) there exists some ε > such thatfor all ε ∈ (0 , ε ) the operator Q ε ( η ε ) satisfies ess spec Q ε ( η ε ) ⊂ [ γε c ρ − /d + , ∞ ) , spec Q ε ( η ε ) = {− ν , } ∪ Λ , where Λ ⊂ [ aε c ρ − /d + , ∞ ) , and ν = ε c ρ − d + (cid:101) ν + o ( ε ) as ε (cid:38) . The first two eigenvalues ν := − ν and ν := 0 are simple with the associated eigenfunctions takingthe form φ i = S ε (cid:101) φ i + o (1) in H k as ε (cid:38) . Spectrum of the linearized augmented Hamiltonian.Lemma 3.15 (Extension of D E c ) . Let { U c } be one of the family of bound states { U A c } , { U A ± c } ,or { U C c } given by Corollaries 2.13, 2.14 or Lemma 2.15 (a) , respectively. Then D E c ( U c ) extendsuniquely to a bounded linear operator H c : X → X ∗ such that D E c ( U c )[ ˙ u, ˙ v ] = (cid:104) H c ˙ u, ˙ v (cid:105) X ∗ × X for all ˙ u, ˙ v ∈ V , and I − H c is self-adjoint on X .Proof. It suffices to consider the diagonal, so let a bound state U c = ( η c , ψ c ) and ˙ u = ( ˙ η, ˙ ψ ) ∈ V begiven. By Lemmas 3.4 and 3.5, we have thatD E c ( U c )[ ˙ u, ˙ u ] = D V aug c ( η c )[ ˙ η, ˙ η ] + (cid:90) R ( T c − S c ) ˙ ηA ( η c )( T c − S c ) ˙ η d x + 2 D ψ D η E c ( U c )[ ˙ η, ˙ ψ ] + D ψ E c ( U c )[ ˙ ψ, ˙ ψ ]= (cid:104) Q c ( η c ) ˙ η, ˙ η (cid:105) X ∗ × X + (cid:90) R ( T c − S c ) ˙ ηA ( η c )( T c − S c ) ˙ η d x + 2 (cid:90) R ˙ ψ (cid:104) D A ( η c ) ˙ η, ψ c (cid:105) d x + 2 c (cid:90) R ˙ η (cid:48) ˙ ψ d x + (cid:90) R ˙ ψA ( η c ) ˙ ψ d x, where we write S c and T c to indicate that these operators are being evaluated at U c . The firstderivative formula in Lemma 2.3 then givesD E c ( U c )[ ˙ u, ˙ u ] = (cid:104) Q c ( η c ) ˙ η, ˙ η (cid:105) X ∗ × X + (cid:90) R ( T c − S c ) ˙ ηA ( η c )( T c − S c ) ˙ η d x + (cid:90) R ˙ ψA ( η c ) ˙ ψ d x + 2 c (cid:90) R ˙ η (cid:48) ˙ ψ d x + 2 (cid:88) ± ρ ± (cid:90) R (cid:18) ( b ± c + c ) (cid:16) G ± ( η c ) − A ( η c ) ˙ ψ (cid:17) (cid:48) ± b ± c A ( η c ) ˙ ψ (cid:19) ˙ η d x, where ( b ± c , b ± c ) is the relative velocity determined by U c via (3.3). Recalling the definitions of S and T in (3.5), this can be expressed quite concisely as:(3.47) D E c ( U c )[ ˙ u, ˙ u ] = (cid:104) Q c ( η c ) ˙ η, ˙ η (cid:105) X ∗ × X + (cid:90) R (cid:16) ( T c − S c ) ˙ η + ˙ ψ (cid:17) A ( η c ) (cid:16) ( T c − S c ) ˙ η + ˙ ψ (cid:17) d x. It is then clear that D E c ( U c ) extend to an element of X ∗ . (cid:3) We can now state and prove the main result of this section, which characterizes the spectrum of E c . It corresponds to [51, Assumption 6]. Theorem 3.16 (Spectrum) . Let { U c } be one of the family of bound states { U A c } , { U A ± c } , or { U C c } given by Corollaries 2.13, 2.14 or Lemma 2.15 (a) , respectively. Then spec I − H c = {− µ c , } ∪ Σ c , NTERNAL WAVE STABILITY 33 where − µ c < is a simple eigenvalue corresponding to a unique eigenvector χ c ; is a simpleeigenvalue generated by T ; and Σ c ⊂ (0 , ∞ ) is bounded uniformly away from .Proof. This follows from the structure of I − H c and a soft analysis argument as in [45, Proposition5.3]. Due either to Theorem 3.12 or Theorem 3.14, the operator Q c ( η c ) + ( α − ν c ) (cid:104) · , φ c (cid:105) φ c + α (cid:104) · , η (cid:48) c (cid:105) η (cid:48) c is positive definite for all α > , where − ν c is the negative eigenvalue of Q c ( η c ) and φ c is thecorresponding eigenfunction. As A ( η c ) is itself positive definite, from (3.47) we obtain the estimate (cid:104) H c u, u (cid:105) X ∗ × X + ( α − ν c ) (cid:104) I − ( φ c , , u (cid:105) X ∗ × X + α (cid:104) I − ( η (cid:48) c , , u (cid:105) X ∗ × X (cid:38) c (cid:107) u (cid:107) X , for all u ∈ X . Thus I − H c is positive definite on a codimension subspace.On the other hand, we know that T (cid:48) (0) U c is in the kernel of H c , and by (3.47) we have that (cid:104) H c u, u (cid:105) X ∗ × X = (cid:104) Q c ( η c ) φ c , φ c (cid:105) X ∗ × X = − ν c < for u = ( φ c , ( S c − T c ) φ c ) . Thus I − H c has a one-dimensional kernel generated by T (cid:48) (0) U c , a one-dimensional negative definitesubspace, and it is positive definite in the orthogonal complement. The claimed spectral propertiesof I − H c are now easily confirmed. (cid:3) Proof of the main results
Finally, in this section we will give the proof of the stability theorems discussed in Section 1. Inorder to state them more concisely, we introduce the following notation. For a fixed bound state U c and radius r > , we define the tubular neighborhoods U X r := { u ∈ O : inf s ∈ R (cid:107) u − T ( s ) U c (cid:107) X < r } , U W r := { u ∈ O ∩ W : inf s ∈ R (cid:107) u − T ( s ) U c (cid:107) W < r } . Similarly, for any
R > , let B W R denote the intersection of O with the ball of radius R centeredat the origin in W . Then U c is said to be conditionally orbitally stable provided that for all r > and R > , there exists r > such that if u : [0 , t ) → B W R is a solution to (2.25) with u (0) ∈ U X r ,then u ( t ) ∈ U X r for all t ∈ [0 , t ) . On the other hand, we say that U c is orbitally unstable providedthat there exists ν > such that, for all < ν < ν there exists initial data in U W ν for which thecorresponding solution exits U W ν in finite time.4.1. Stability of uniform flows.
We begin with the simpler case of the trivial solution U c = (0 , ,corresponding to a laminar flow with (the same) constant purely horizontal velocity in each layer.For ( β, λ ) in Region B, we then have by Lemma 3.7 that I − H c is positive definite. Let us now stateand prove a rigorous version of Theorem 1.3. Because U c = 0 , the tubular neighborhoods abovesimply become balls in the appropriate spaces, and hence conditional orbital stability is equivalentto conditional stability. Theorem 4.1 (Stability of uniform flows) . Let U c = (0 , be the trivial bound state for the internalwave problem (2.25) with wave speed c ∈ R . Then U c is conditionally stable if the corresponding ( β, λ ) lies in Region B.Proof. Because E c is C ∞ ( V ; R ) , E c ( U c ) = 0 , and D E c ( U c ) = 0 , Taylor expanding it at U c gives E c ( u ) = 12 (cid:104) H c u, u (cid:105) + O ( (cid:107) u (cid:107) V ) . For ( β, λ ) in Region B, we have by Lemmas 3.7 and 3.15 that I − H c is positive definite on X . Onthe other hand, the cubic term above can be controlled via Lemma 2.2: (cid:107) u (cid:107) V (cid:46) (cid:107) u (cid:107) − θ W (cid:107) u (cid:107) θ X ≤ r θ R − θ (cid:107) u (cid:107) X for all u ∈ U X r ∩ B W R . Thus, for r > sufficiently small, it holds that(4.1) E c ( u ) ≥ α (cid:107) u (cid:107) X for all u ∈ U X r ∩ B W R , for some α = α ( r, R ) > .Now, seeking a contradiction, suppose that U c is not conditionally stable. Thus there exists R > , r > , and a sequence of initial data { u n } ⊂ O ∩ W with u n → in X but for which thecorresponding solution u n : [0 , t n ) → B W R exits U X r in finite time: (cid:107) u n ( τ n ) (cid:107) X = r for some τ n ∈ (0 , t n ) .Let τ n be the first such time and, if necessary, shrink r so that (4.1) holds. Together with theconservation of energy and momentum, this ensures that(4.2) E c ( u n ) = E c ( u n ( τ n )) ≥ αr for all n ≥ . Because { u n } ⊂ B W R and u n → in X , Lemma 2.2 forces u n → in V . But then, the continuityof E and P would imply that E c ( u n ) also vanishes in the limit. As this is in obvious contradictionwith (4.2), the proof is complete. (cid:3) Stability for strong surface tension.
Next, we turn to the more complicated situation wherethe wave in question is small-amplitude but nontrivial. Consider first the strong surface tension casecorresponding to the waves in Region A. In Theorem 3.16, it was shown that I − H c has a negativedirection in this regime, and so we will use the energy-momentum approach to show stability. Havinglaid the groundwork for this argument in the previous sections, we are prepared to state and prove aprecise version of Theorem 1.1. Theorem 4.2 (Stability for strong surface tension) . For all c such that < λ c − λ (cid:28) , the boundstates U A c and U A ± c given by Corollaries 2.13 and 2.14 are conditionally orbitally stable.Proof. Let U c stand for both U A c and U A ± c , as the first stage of the proof is identical in either case.In Section 2, we confirmed that Assumptions – of [51] hold, and Assumption was verified inTheorem 3.16. By [51, Theorem 2.4], to prove that U c ∗ is conditionally orbitally stable we need onlyshow that d (cid:48)(cid:48) ( c ∗ ) > , where d is moment of instability defined by(4.3) d ( c ) := E c ( U c ) = E ( U c ) − cP ( U c ) . Because U c is a critical point of E c , differentiating the above equation gives(4.4) d (cid:48) ( c ) = − P ( U c ) . Thus we must confirm that c (cid:55)→ − P ( U c ) is strictly increasing at c = c ∗ .The definition of the momentum (2.29) and kinematic condition (3.1) yield the explicit formula d (cid:48) ( c ) = (cid:90) R η (cid:48) c ψ c d x = c (cid:90) R η c ∂ x A ( η c ) − η (cid:48) c d x. As in Section 3.2, we will exploit a long-wave rescaling to analyze this quantity. Recycling notation,let us redefine the scaling operator to be(4.5) S c f := f (cid:18) ε c · d + √ β c − β (cid:19) , where ε c = ε A c and β c are given by (2.37). Likewise, the asymptotics for the free surface profileestablished in (2.35) and (2.36) permits us to write η c =: ε mc d + S c ( (cid:101) η + (cid:101) r c ) for (cid:101) r c = O ( ε c ) in H k , NTERNAL WAVE STABILITY 35 with m = 2 for U A c and m = 1 for U A ± c . Using the rescaling, we compute that d (cid:48) ( c ) = cε mc d (cid:90) R ( S c ( (cid:101) η + (cid:101) r c )) ∂ x A ( η c ) − ∂ x S c ( (cid:101) η + (cid:101) r c ) d x = cε m − c d (cid:112) β c − β (cid:90) R ( (cid:101) η + (cid:101) r c ) S − c ∂ x A ( η c ) − ∂ x S c ( (cid:101) η + (cid:101) r c ) d x = cε m − c d (cid:112) β c − β (cid:88) ± ρ ± (cid:90) R ( (cid:101) η + (cid:101) r c ) S − c ∂ x G ± ( η c ) − ∂ x S c ( (cid:101) η + (cid:101) r c ) d x, where the last line follows from (2.21). Similar to (3.27), let us define (cid:102) M ± c ( η c ) := d + S − c ∂ x G ± ( η c ) − ∂ x S c . Arguing as in Lemma 3.9, we then find that (cid:13)(cid:13)(cid:13)(cid:13) (cid:102) M ± c (0) + d + d ± (cid:13)(cid:13)(cid:13)(cid:13) Lin( H ,L ) (cid:46) ε c , (cid:13)(cid:13)(cid:13) (cid:102) M ± c ( η c ) − (cid:102) M ± c (0) (cid:13)(cid:13)(cid:13) Lin( H ,L ) (cid:46) ε mc , and hence(4.6) d (cid:48) ( c ) = cε m − c d (cid:112) β c − β (cid:88) ± ρ ± (cid:90) R (cid:101) η (cid:102) M ± c (0) (cid:101) η d x + O ( ε m − c ) in C ( I )= − cε m − c d (cid:112) β c − β (cid:88) ± ρ ± d + d ± (cid:90) R (cid:101) η d x + O ( ε m − c ) in C ( I ) , where recall that I is a sufficiently small interval containing c ∗ .Now, observe that ε c , β c > , and from (2.37), c (cid:55)→ ε c and c (cid:55)→ cβ c sgn c are both positive and (cid:26) strictly decreasing for c > , andstrictly increasing for c < , Therefore c (cid:55)→ − cε m − c ( β c − β ) / is strictly increasing. This completes the proof for the family { U A c } , as (cid:101) η is independent of c in that case.The argument for { U A ± c } is only slightly more complicated. Recall that by (2.36),(4.7) (cid:101) η = (cid:101) η A ± c ( x ) = 1 κ c ± (cid:112) κ c + 4( (cid:37) + h ) cosh x , with κ c = κ A c defined as in Corollary 2.14. Since we are in fact computing (cid:82) R (cid:101) η d x , it is sufficient toassume that κ c > . Then, clearly (cid:101) η A+ c > and c (cid:55)→ κ c sgn c is increasing, so we again have by (4.6)and the argument in the previous paragraph that d (cid:48) is strictly increasing. Finally, (cid:101) η A − c is a wave ofdepression and an explicit computation using (4.7) gives (cid:90) R ( (cid:101) η A − c ) d x = κ c tan − (cid:18) κ c + √ κ c +4( (cid:37) + h )4( (cid:37) + h ) (cid:19) (cid:37) + h ) / − (cid:37) + h ) . It is easily seen that the right-hand side above is strictly increasing in c for c > and strictlydecreasing for c < . The proof is therefore complete. (cid:3) Stability for near critical surface tension.
Consider now the families of bound states { U C c } that correspond to traveling waves in Region C. Recall from Section 2.5, that to leading order, thecorresponding free surface profiles are rescalings of the family of primary homoclinic orbits { Z δ } ofthe ODEs (2.39). To unify the presentation, we will write Z c as shorthand for Z δ C c .The next theorem shows that under the hypothesis of Theorem 3.14, the orbital stability/instabilityof these waves can be inferred purely from properties of the primary homoclinic orbits. Theorem 4.3 (Stability for critical surface tension) . Consider the family of traveling waves { U C c } given in Lemma 2.15 (a) and assume that the hypothesis of Theorem 3.14 holds. For all c ∗ with < λ c ∗ − λ (cid:28) , the corresponding wave is conditionally orbitally stable provided that the function (4.8) c (cid:55)→ sgn c (cid:90) R Z c d x is strictly increasing at c ∗ , and it is orbitally unstable if this function is strictly decreasing there.Proof. Throughout the argument, we abbreviate { U c } for { U C c } and ε c = ε C c . We have alreadyproved in Theorem 3.16 that the spectral hypothesis on I − H c in [51, Assumption 6] holds. As in theprevious subsection, we may therefore apply [51, Theorem 2.4] to conclude that U c ∗ is conditionallyorbitally stable provided that d (cid:48)(cid:48) ( c ∗ ) > , where d is the moment of instability (4.3). On the otherhand, because the Cauchy problem is locally well-posed, [51, Assumption 7] is satisfied, and so [51,Theorem 2.6] tells us that U c ∗ is orbitally unstable if d (cid:48)(cid:48) ( c ∗ ) < .From Lemma 2.15, we know that free surface profile takes the form η c = ε c d + S c ( Z c + (cid:101) r c ) with (cid:101) r c = O ( ε c ) in H k as ε (cid:38) , where we have redefined the scaling operator to be S c f := f ( ε · /d + ) . The same argument as in theproof of Theorem 4.2 reveals that d (cid:48) ( c ) = − cε c d γ (cid:88) ± ρ ± d + d ± (cid:90) R Z c d x + O ( ε c ) in C ( I ) . From the definition of ε c in (2.41), c (cid:55)→ ε c and c (cid:55)→ cε c sgn c are both positive and (cid:26) strictly decreasing for c > , andstrictly increasing for c < , Thus c (cid:55)→ − cε c is strictly increasing. Therefore d (cid:48) ( c ) is strictly increasing at c ∗ when (4.8) issatisfied. (cid:3) We remark that (4.8) is stated in terms of the wave speed c , but to compare it to results ondispersive model equations of Kawahara type (1.9) it is natural to consider the related function δ (cid:55)→ (cid:82) Z δ d x . Looking carefully at its definition in (2.41), we see that c (cid:55)→ δ C c can be both increasingor decreasing depending on the various physical parameters. Acknowledgments
The research of RMC is supported in part by the NSF through DMS-1907584. The research ofSW is supported in part by the NSF through DMS-1812436. The authors would also like to thanksDag Nilsson for enlightening communications regarding the existence theory in Section 2.5, andDaniel Sinambela for close readings of earlier versions of the manuscript.
Appendix A. Elementary identities
Proof of Lemma 3.2.
As in the proof of Lemma 2.3, we start by considering the correspondingformula for A ( η ) − . Recalling (2.21), we see that D ( A ( η ) − )[ ˙ η, ˙ η ] = (cid:88) ± ρ ± D ( G ± ( η ) − )[ ˙ η, ˙ η ]= − (cid:88) ± ρ ± G ± ( η ) − (cid:0) D G ± ( η )[ ˙ η, ˙ η ] − D G ± ( η )[ ˙ η ] G ± ( η ) − D G ± ( η )[ ˙ η ] (cid:1) G ± ( η ) − . On the other hand, we have the elementary identityD A ( η )[ ˙ η, ˙ η ] = − A ( η ) D ( A ( η ) − )[ ˙ η, ˙ η ] A ( η ) + 2 D A ( η )[ ˙ η ] A ( η ) − D A ( η )[ ˙ η ] . (A.1) NTERNAL WAVE STABILITY 37
Together, these will furnish a representation formula for the second variation of A ( η ) − once we havefully expanded these expressions using (2.18) and (3.7).Consider each of the terms on the right-hand side of (A.1). For the first, we have − (cid:90) R ψA ( η ) D ( A ( η ) − )[ ˙ η, ˙ η ] A ( η ) ψ d x = (cid:88) ± ρ ± (cid:90) R θ ± D G ± ( η )[ ˙ η, ˙ η ] θ ± d x − (cid:88) ± ρ ± (cid:90) R θ ± D G ± ( η )[ ˙ η ] G ± ( η ) − D G ± ( η )[ ˙ η ] θ ± d x, where recall that θ ± = θ ± ( η, ψ ) is given by (3.10). Throughout the remainder of the proof, a ± i willalways be evaluated at ( η, θ ± ) , so we suppress the arguments for readability. By the first variation(2.18) and second variation (3.7) formulas for G ± ( η ) , this becomes − (cid:90) R ψA ( η ) D ( A ( η ) − )[ ˙ η, ˙ η ] A ( η ) ψ d x = (cid:88) ± ρ ± (cid:90) R (cid:0) a ± ˙ η + 2 a ± ˙ ηG ± ( η ) (cid:0) a ± ˙ η (cid:1)(cid:1) d x − (cid:88) ± ρ ± (cid:90) R a ± (cid:0) G ± ( η ) − D G ± ( η )[ ˙ η ] θ ± (cid:1) (cid:48) ˙ η d x − (cid:88) ± ρ ± (cid:90) R a ± ( D G ± ( η )[ ˙ η ] θ ± ) ˙ η d x = (cid:88) ± ρ ± (cid:90) R (cid:0) a ± ˙ η + 2 a ± ˙ ηG ± ( η ) (cid:0) a ± ˙ η (cid:1)(cid:1) d x − (cid:88) ± ρ ± (cid:90) R L ± [ ˙ η ] D G ± ( η )[ ˙ η ] θ ± d x, for the linear operator L ± given by (3.11). Using (2.18) once more allows us to simplify this to − (cid:90) R ψA ( η ) D ( A ( η ) − )[ ˙ η, ˙ η ] A ( η ) ψ d x = (cid:88) ± ρ ± (cid:90) R (cid:0) a ± ˙ η + 2 a ± G ± ( η ) (cid:0) a ± ˙ η (cid:1)(cid:1) ˙ η d x − (cid:88) ± ρ ± (cid:90) R (cid:0) a ± L ± [ ˙ η ] (cid:48) + a ± G ± ( η ) L ± [ ˙ η ] (cid:1) ˙ η d x. So finally we have(A.2) − (cid:90) R ψA ( η ) D ( A ( η ) − )[ ˙ η, ˙ η ] A ( η ) ψ d x = (cid:90) R (cid:16) a ˙ η + 2 (cid:88) ± ρ ± a ± G ± ( η ) (cid:0) a ± ˙ η (cid:1) − M ˙ η (cid:17) ˙ η d x where recall a = a ( η, ψ ) and M = M ( η, ψ ) were defined in (3.10) and (3.12), respectively.Likewise, the second in term on the right-hand side of (A.1) can be treated as follows. Using(2.20), we calculate that (cid:90) R ψ D A ( η )[ ˙ η ] A ( η ) − D A ( η )[ ˙ η ] ψ d x = (cid:88) ± ρ ± (cid:90) R (cid:16) a ± (cid:0) G ± ( η ) − D A ( η )[ ˙ η ] ψ (cid:1) (cid:48) + a ± A ( η ) D A ( η )[ ˙ η ] ψ (cid:17) ˙ η d x = (cid:88) ± ρ ± (cid:90) R L ± [ ˙ η ] D A ( η )[ ˙ η ] ψ d x = (cid:90) R L [ ˙ η ] D A ( η )[ ˙ η ] ψ d x. Applying (2.20) once more then yields (cid:90) R ψ D A ( η )[ ˙ η ] A ( η ) − D A ( η )[ ˙ η ] ψ d x = (cid:88) ± ρ ± (cid:90) R (cid:16) a ± (cid:0) A ( η ) G ± ( η ) − L [ ˙ η ] (cid:1) (cid:48) + a ± A ( η ) L [ ˙ η ] (cid:17) ˙ η d x = (cid:90) R ˙ η N ˙ η d x, (A.3)with N = N ( η, ψ ) defined in (3.13). Combining this with (A.1) and (A.2) gives the formula (3.9),completing the proof. (cid:3) References [1]
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