Stability of periodic waves in Hamiltonian PDEs
aa r X i v : . [ m a t h . A P ] D ec Stability of periodic waves in Hamiltonian PDEs
Sylvie Benzoni-Gavage ∗ , Pascal Noble † , and L.Miguel Rodrigues ‡ October 10, 2018
Abstract
Partial differential equations endowed with a Hamiltonian structure,like the Korteweg–de Vries equation and many other more or less classicalmodels, are known to admit rich families of periodic travelling waves. Thestability theory for these waves is still in its infancy though. The issuehas been tackled by various means. Of course, it is always possible toaddress stability from the spectral point of view. However, the link withnonlinear stability - in fact, orbital stability, since we are dealing withspace-invariant problems - , is far from being straightforward when thebest spectral stability we can expect is a neutral one. Indeed, because ofthe Hamiltonian structure, the spectrum of the linearized equations can-not be bounded away from the imaginary axis, even if we manage to dealwith the point zero, which is always present because of space invariance.Some other means make a crucial use of the underlying structure. Thisis clearly the case for the variational approach, which basically uses theHamiltonian - or more precisely, a constrained functional associated withthe Hamiltonian and with other conserved quantities - as a Lyapunovfunction. When it works, it is very powerful, since it gives a straightpath to orbital stability. An alternative is the modulational approach,following the ideas developed by Whitham almost fifty years ago. Themain purpose here is to point out a few results, for KdV-like equationsand systems, that make the connection between these three approaches:spectral, variational, and modulational.
Acknowledgments
This work has been partly supported by the EuropeanResearch Council ERC Starting Grant 2009, project 239983- NuSiKiMo, and theAgence Nationale de la Recherche, JCJC project Shallow Water Equations forComplex Fluids 2009-2103. This paper has been prepared for the Proceedings of
Journ´ees ´Equations aux D´eriv´ees Partielles , GDR CNRS 2434, Biarritz 2013.
Keywords periodic travelling wave, variational stability, spectral stability,modulational stability ∗ [email protected] † [email protected] ‡ [email protected] lassification Sound and light are manifestations of periodic waves, even though they arehardly perceived as waves in daily life. Perhaps the most famous, clearly visi-ble periodic waves are those propagating at the surface of water, named afterGeorge Gabriel Stokes. In real-world situations, periodic water waves can beformed for instance by ships. Their two main features are non-linearity and dis-persion , which imply that their velocity depends on both their amplitude andtheir wavelength. However, it was observed in a celebrated work [3] that theso-called Stokes waves were not so easy to create in lab experiments. At firstpuzzled by this problem, Benjamin and Feir exhibited a threshold for the ratioof depth over wave length above which small amplitude Stokes waves becomeunstable.If the Stokes waves are an archetype of nonlinear dispersive waves, the un-derlying - water wave - equations are quite complicated. The purpose of thistalk was to give an overview of stability theory for a wide range of nonlineardispersive waves, of possibly arbitrary amplitude, arising as solutions of PDEsendowed with a ‘nice’ algebraic structure. This has been a renewed, active fieldin the last decade, with still a number of open questions even in one spacedimension. By contrast, the theory is much more advanced regarding solitarywaves, which may be viewed as a limiting case of periodic waves - namely, whentheir wavelength goes to infinity.We restrict to one-dimensional issues in what follows. In mathematicalphysics, there are a number of model equations supporting nonlinear dispersivewaves. The most classical ones are known as the
Non-Linear Wave equation(NLW) ∂ t χ − ∂ x χ + v ( χ ) = 0 , the (generalized) Boussinesq equation(B) ∂ t φ − ∂ x ( w ( φ ) ∓ ∂ x φ ) = 0 , the (generalized) Korteweg-de Vries equation(KdV) ∂ t v + ∂ x p ( v ) = − ∂ x v , and the Non-Linear Schr¨odinger equation(NLS) i∂ t ψ + ∂ x ψ = ψ g ( | ψ | ) . It is on purpose that we have chosen to write non-linear terms in their mostgeneral form here above - observe that nonlinearities are written as v ( χ ) in(NLW), w ( φ ) in (B), p ( v ) in (KdV), and ψ g ( | ψ | ) in (NLS). As a matter offact, we shall refrain from invoking integrability arguments, which only work forsome specific nonlinearities. Nevertheless, a common feature of these equations2s that they are endowed with a Hamiltonian structure. Indeed, they can all bewritten in the abstract form ∂ t U = J ( E H [ U ]) , (1)where the unknown U takes values in R N ( N = 1 for (KdV), N = 2 for(B), (NLS), N = 3 for (NLW)), J is a skew-adjoint differential operator,and E H denotes the variational derivative of H , whose α -th component ( α ∈{ , . . . , N } ) merely reads as follows when H = H ( U , U x ),( E H [ U ]) α := ∂ H ∂U α ( U , U x ) − D x (cid:18) ∂ H ∂U α,x ( U , U x ) (cid:19) . Here above, D x stands for the total derivative. More explicitly, this means thatD x (cid:18) ∂ H ∂U α,x ( U , U x ) (cid:19) = ∂ H ( U , U x ) ∂U β ∂U α,x U β,x + ∂ H ( U , U x ) ∂U β,x ∂U α,x U β,xx , where we have used Einstein’s convention of summation over repeated indices.Another convention is that square brackets [ · ] signal a function of not only thedependent variable U but also of its derivatives U x , U xx , . . . (For instance,we shall either write H ( U , U x ) or H [ U ].) A motivation for addressing thestability of periodic waves in such an abstract setting is to make the most ofalgebra, irrespective of the model under consideration. However, we do have aspecific model in mind, namely the Euler–Korteweg system, which admits twodifferent formulations depending on whether we choose Eulerian coordinates,(EKE) ( ∂ t ρ + ∂ x ( ρu ) = 0 ,∂ t u + u∂ x u + ∂ x ( E ρ E ) = 0 , E = E ( ρ, ρ x ) , or Lagrangian coordinates,(EKL) ( ∂ t v = ∂ y u ,∂ t u = ∂ y ( E v `e ) , `e = `e ( v, v y ) , both fitting the abstract framework in (1). (For details on all these equations,see Table in Appendix.) This is not that a specific model though. Whatwe call the Euler–Korteweg system comprises many models of mathematicalphysics, including the Boussinesq equation for water waves, as well as (NLS)after Madelung’s transformation, see for instance [4] for more details.In the literature on Hamiltonian PDEs, the distinction is often made between‘NLS-like equations’, in which J is merely a real skew-symmetric matrix, and‘KdV-like equations’, in which J = B ∂ x with B a real symmetric matrix. Thisdistinction is to some extent artificial, since for instance (NLS) can be writtenas a special case of the KdV-like system (EKE), and on the contrary (EKE)can take the form of a NLS-like system if the hydrodynamic potential is toreplace the velocity u as a dependent variable. However, there should be a most‘natural’ formulation for each equation or system.3rom now on, we concentrate on KdV-like equations, and assume that J = B ∂ x with B a nonsingular, symmetric matrix. In this case, (1) is itself a systemof conservation laws, which reads ∂ t U = ∂ x ( B E H [ U ]) , (2)and turns out to admit the additional, scalar conservation law ∂ t Q ( U ) = ∂ x ( S [ U ]) (3)with Q ( U ) := U · B − U , S [ U ] := U · E H [ U ] + L H [ U ] , L H [ U ] := U α,x ∂ H ∂U α,x ( U , U x ) − H ( U , U x ) . The dots · in the definitions of Q and S are for the ‘canonical’ inner product U · V = U α V α in R N . The letter L stands for the ‘Legendre transform’ (even thoughit is considered in the original variables ( U , U x )). Equation (3) is satisfied alongany smooth solution of (1). Notice that for any (smooth) function U , ∂ x U = ∂ x ( B E Q [ U ]) . (4)Viewed as ∂ x U = J ( E Q [ U ]), this relation reveals that the (local) conserva-tion law (3) for Q ( U ) is associated with the invariance of (2) under spatialtranslations. Any such quantity has been called an impulse by Benjamin [2].Of course there is also a conservation law associated with the invariance of (1)under time translations, which is nothing but the (local) conservation law forthe Hamiltonian ∂ t H ( U , U x ) = ∂ x (cid:16) E H [ U ] · B E H [ U ] + ∇ U x H [ U ] · D x ( E H [ U ]) (cid:17) . (5)However, this rather complicated conservation law will play a much less promi-nent role than (3) in what follows.For a travelling wave U = U ( x − ct ) of speed c to be solution to (1), onemust have by (4) that ∂ x ( E ( H + c Q )[ U ]) = 0 , or equivalently, there must exist λ ∈ R N such that E ( H + c Q )[ U ] + λ = 0 . (6)This is nothing but the Euler-Lagrange equation associated with the Lagrangian L = L ( U , U x ; c, λ ) := H ( U , U x ) + c Q ( U ) + λ · U . As is well-known, an Euler-Lagrange equation for a Lagrangian L admits L L -the ‘Legendre transform’ of L - as a first integral. Unsurprisingly, this first which also exist for NLS-like equations, but are no longer algebraic and depend on U x ,see Table in Appendix. S + c Q , a quantity that is clearly constant alongthe travelling wave, thanks to (3). The reader may easily check indeed that L L [ U ] = S [ U ] + c Q [ U ]as soon as (6) holds true. Therefore, a full set of equations for the travellingprofile U consists of (6) together with L L [ U ] = µ , (7)where µ is a constant of integration. Recalling that L depends on ( c, λ ), wesee that a travelling profile U depends on ( c, λ , µ ) ∈ R N +2 , which ‘generically’makes the set of profiles an ( N + 2)-dimensional manifold. This is up to trans-lations of course, because any translated version x U ( x + s ) (for an arbitrary s ∈ R ) of U still solves (6)-(7).In practice, the existence of periodic waves is not straightforward. However,it almost becomes so if N = 1 or 2, under a few assumptions that are metby all our KdV-like equations (namely, (KdV) itself, (EKE), and (EKL)). Thesimplest case is N = 1, with the dependent variable U being reduced to a scalarvariable v , and H = E ( v, v x ) , ∂ H ∂v x = ∂ E ∂v x =: κ ( v ) > . (This is a slight generalization of what happens with the usual KdV-equation,in which κ is constant.) A little more complicated case is with N = 2, with thedependent variable U = ( v, u ), and H = H ( v, u, v x ) = E ( v, v x ) + T ( v, u ) , (8)such that ∂ H ∂v x = ∂ E ∂v x =: κ ( v ) > , ∂ H ∂u = ∂ T ∂u =: T ( v ) > . (9) B − = (cid:18) a bb (cid:19) , b = 0 . (10)(These assumptions are met by both (EKE) and (EKL).) In this way, we mayeliminate u from the profile equations (6) and receive a single, second order ODEin v , which also inherits a Hamiltonian structure, and is therefore completelyintegrable . The reader might want to see this equation. Otherwise, they mayskip what follows and go straight to the end of this section.The second component in (6) reads indeed T ( v ) u + ∂ u T ( v,
0) + c v b + λ = 0 , where λ is the second component of λ . Since T ( v ) is nonzero, this gives u = f ( v ; c, λ ) := − T ( v ) − ( ∂ u T ( v,
0) + c v b + λ ) .
5y plugging this expression in (7), we arrive at L E [ v ] − T ( v, f ( v ; c, λ )) − c (cid:16) av + vbf ( v ; c, λ ) (cid:17) − λ v − λ f ( v ; c, λ ) = µ . Despite its terrible aspect, this equation is merely of the form12 κ ( v ) v x + W ( v ; c, λ ) = µ , (11)if E is really quadratic in v x ( i.e. if ∂ v x E ( v,
0) = 0). We obtain a similarone in the case N = 1 with H = E ( v, v x ). Eq. (11) can be viewed as anintegrated version of the Euler–Lagrange ODE, E ℓ = 0, associated with the‘reduced’ Lagrangian ℓ := 12 κ ( v ) v x − W ( v ; c, λ ) . Incidentally, E ℓ = 0 admits as a first integral the ‘reduced’ Hamiltonian h := 12 κ ( v ) v x + W ( v ; c, λ ) . We thus find families of periodic orbits parametrized by µ around any localminimum of the potential W ( · ; c, λ ). In case W ( · ; c, λ ) is a double-well potential,which is what happens with the famous van der Waals/Cahn–Hilliard/Wilsonenergies, a same parameter µ can clearly be associated with two different orbits.In other words, the whole set of periodic orbits is not made of a single graph overthe set of parameters ( µ, λ , c ). Nevertheless, each family of periodic orbits canbe parametrized by ( µ, c, λ ), as long as the wells of W ( · ; c, λ ) remain distinct.Going back to the more comfortable general setting, let us just assume thatthere exist open sets of parameters ( µ, λ , c ) for which (6)-(7) have a uniqueperiodic solution up to translations. Note that the set of solitary wave profilesmay be viewed as a co-dimension one boundary of periodic profiles. Indeed,for a solitary wave profile, once λ has been prescribed by the endstate U ∞ =( v ∞ , u ∞ ), λ = − ∇ U ( H + c Q )( U ∞ , , the constant of integration µ is given by µ = − H ( v ∞ , u ∞ , − c Q ( U ∞ ) − λ · U ∞ . We now aim at investigating the stability of periodic travelling waves U = U ( x − ct ). For this purpose, some global, stringent assumptions — for instancequadraticity in v x and u — may often be relaxed to suitable, local invertibilityassumptions. Let us consider a periodic travelling wave U = U ( x − ct ) solution to (1). In otherwords, we assume that U is a periodic solution to (6)-(7), and denote by Ξ its6eriod . The latter is supposed to be uniquely determined, say in the vicinity ofa reference profile, by the parameters ( µ, λ , c ). As to the profile U , it can onlybe unique up to translations. Thus, we may assume without loss of generalitythat v x (0) = 0. This choice will play a role in subsequent calculations. Let usnow review a series of related notions and tools. By the Euler–Lagrange equation in (6), U is a critical point of the functional F ( c, λ ,µ ) : U Z Ξ0 ( H ( U , U x ) + c Q ( U ) + λ · U + µ ) d x . (At this point, the µ term does not play any role but it will come into play lateron.) Would in additionΘ( µ, λ , c ) := F ( c, λ ,µ ) [ U ] = Z Ξ0 ( H ( U , U x ) + c Q ( U ) + λ · U + µ ) d x be a (locally) minimal value of F ( c, λ ,µ ) , it would be natural to use this functionalas a Lyapunov function in order to show the stability of U . This would require,though, that its Hessian, A := Hess ( H + c Q )[ U ]be a positive differential operator. (Of course A depends on the parameters( µ, λ , c ) but we omit to write them in order to keep the notation simple.) Thiswe would call variational stability. However, there is no hope that it be thecase. A first reason is, by differentiating (6) with respect to c , we readily seethat A U x = 0. Hence A has a nontrivial kernel on L ( R / Ξ Z ), containing atleast U x , as is always the case with space-invariant problems. An even worseobservation is that, by a Sturm–Liouville argument applied to the second orderODE satisfied by v , the equality A U x = 0 certainly implies that A has anegative eigenvalue (see Appendix for more details). Nevertheless, what we canhope for is constrained variational stability. Indeed, knowing that U and Q ( U )are conserved quantities, it can be that the values of F ( c, λ ,µ ) which are lowerthan Θ( µ, λ , c ) are not seen on the manifold C := { U ; R Ξ0 Q ( U ) d x = R Ξ0 Q ( U ) d x , R Ξ0 U d x = R Ξ0 U d x } . By ‘not seen’ we mean an infinite-dimensional analogue of what happens forinstance with the indefinite function ( x, y ) y − x , which does have a (local)minimum along any curve lying in { (0 , } ∪ { ( x, y ) ; | x | < | y |} . Determin-ing whether C is located in the ‘good’ region amounts to identifying suitable Please note that this is a spatial period. We refrain from using the word ‘wavelength’here in order to prevent the reader from thinking U as a harmonic wave. It can be a cnoidalwave, or any kind of periodic wave. A is nonnegative on the tangentspace T U C , a necessary condition for the functional F ( c, λ ,µ ) to be minimizedat U along C . Then we may speak of constrained variational stability de-spite the translation-invariance problem, that is, even though U is not a strictminimizer. Indeed, as observed in earlier work on solitary waves [13, Lemma3.2], any U close to U admits by the implicit function theorem a translate x U ( x + s ( U )) such that U ( · + s ( U )) − U is orthogonal to U x with respectto the L inner product. This argument clearly paves the way towards orbital stability. As a matter of fact, by reasoning as in [13, Theorem 3.5] with anappropriate choice of a function space H ⊂ L ( R / Ξ Z ) in which we would havea flow map U (0) U ( t ) for (1), we might prove that ∀ ε > , ∃ δ > k U (0) − U k H ≤ δ ⇒ ∀ t ≥ , inf s ∈ R k U ( t ) − U ( · + s ) k H ≤ ε . This would mean orbital stability of U with respect to co-periodic perturba-tions ( H being made of Ξ-periodic functions). Possibly redefining F ( c, λ ,µ ) asan integral over an interval of length n Ξ for an integer n ≥
2, we might alsoprove orbital stability with respect to multiply periodic perturbations, that is in L ( R /n Ξ Z ). Note however that this would require a more delicate count of sig-natures [11], because the negative spectrum of A grows bigger when n increases(again by a Sturm–Liouville argument). As to ‘localized’ perturbations, there isno obvious definition of a functional that would play the role of F ( c, λ ,µ ) . Thisdiffers from the case of solitary waves, for which M ( U ∞ ,c ) : U Z ∞−∞ ( H ( U , U x ) + c Q ( U ) + λ · U + µ ) d x does the job. If U = U ( x − ct ) is a solitary wave homoclinic to U ∞ , theintegral M ( U ∞ , c ) := M ( U ∞ ,c ) ( U ) has been known as the Boussinesq momentof instability , and the Grillakis-Shatah-Strauss criterion requires that ∂ M ∂c =: M cc > M cc thatplays a role in the solitary wave stability is not difficult to see, as soon as wehave in mind the following crucial relations, A U c = ∇ Q ( U ∞ ) − ∇ Q ( U ) =: q , M cc = −h q · U c i L , obtained by differentiating the profile equation (6) with respect to c at fixed U ∞ , and of course also M . Assuming that M cc is nonzero, we thus see thatany U ∈ D ( A ) can be decomposed in a unique way as U = a U c + V with h q · V i L = 0, and h A U · U i L = − a M cc + h A V · V i L .
8n this identity we see that the negative signature n ( A ) of A equals the one n ( A | q ⊥ ) of A | q ⊥ if M cc <
0, whereas n ( A ) = n ( A | q ⊥ ) + 1if M cc >
0. In the latter situation, if it is true that A has a single negativeeigenvalue, we find that A | q ⊥ has no negative spectrum, hence constrained vari-ational stability. (The proof of orbital stability then follows by a contradictionargument [13, 8].) On the other hand, A | q ⊥ does have negative spectrum if M cc <
0, hence constrained variational instability. (The proof in [13] that thisimplies orbital instability is trickier, and does not work if we cannot assure thatthere is a negative direction y of A | q ⊥ in the range of J , which is equivalentto requiring that R + ∞−∞ y d x = 0 if J = B ∂ x . This issue was fixed in [8] for(KdV).)Let us go back to periodic waves. The functional F ( c, λ ,µ ) defined at thebeginning of this section turns out to be a ubiquitous tool for the stabilityanalysis of the periodic travelling waves U = U ( x − ct ) defined by (6)-(7). Weshall repeatedly meet its second variational derivative, A = Hess ( H + c Q )[ U ],which depends not only on c but also on ( λ , µ ) through the profile U and whosespectrum undoubtedly plays a crucial role in the stability or instability of U .In addition, the value of F ( c, λ ,µ ) at U , which we have denoted by Θ( µ, λ , c ),and the variations of Θ with respect to ( c, λ , µ ) show up in stability conditionsfrom both the spectral and modulational points of view. A widely used approach to stability of equilibria consists in linearizing aboutthese equilibria. Even though periodic waves U = U ( x − ct ) are not genuineequilibria, they can be changed into stationary solutions by making a change offrame. Indeed, in a frame moving with speed c , Eq. (2) becomes ∂ t U − c∂ x U = ∂ x ( B E H [ U ]) , or equivalently, ∂ t U = B ∂ x ( E ( H + c Q )[ U ]) , which admits U = U ( x ) as special solutions. Linearizing about U we receivethe system ∂ t U = B ∂ x ( A U ) , where we recognize A = Hess ( H + c Q )[ U ]. Therefore, the linearized stabilityof U should be encoded by the spectrum of A = J A with J = B ∂ x . Bydefinition, U will be said to be spectrally stable if the operator A has no spectrumin the right-half plane. Note that, since J is skew-adjoint and A is self-adjoint- and both are real-valued -, possible eigenvalues of A arise as quadruplets( τ, τ , − τ, − τ ). This means that any eigenvalue outside the imaginary axis wouldimply instability. Furthermore, according to [17, Theorem 3.1], the number of9igenvalues of A in the left-half plane controls, in some sense, the number ofunstable eigenvalues of A . Recalling that A has at least one negative eigenvalue,there is room for (at least) one unstable eigenvalue of A .These considerations are rather loose actually, because the spectrum of adifferential operator depends on the chosen functional framework. We may lookat the differential operator A as an unbounded operator on L ( R / Ξ Z ), in whichcase its spectrum is entirely made of isolated eigenvalues. These concern what isusually called co-periodic spectral stability. We may widen the class of possibleperturbations and consider A as an unbounded operator on L ( R /n Ξ Z ) with n any integer greater than one. Finally, we may consider ‘localized’ perturbationsby looking at A as an unbounded operator on L ( R ). As was shown by Gardner[12], the spectrum of A on L ( R ) is made of a collection of closed curves of so-called ν -eigenvalues. For any ν ∈ R / π Z , a ν -eigenvalue is an eigenvalue of theoperator A ν := A ( ∂ x + iν/ Ξ) on L ( R / Ξ Z ). These definitions are motivatedby the equivalence, which holds for all τ ∈ C ,( AU = τ U , U ( · + Ξ) = e iν U ) ⇔ ( A ν U ν = τ U ν , U ν ( · + Ξ) = U ν ) , where we have introduced the additional notation U ν : x U ν ( x ) = e − iνx/ Ξ U ( x ) . All this is linked to the Floquet theory of ODEs with periodic coefficients, and weshall refer to ν as a Floquet exponent . Furthermore, there is a tool encoding allkinds of spectral stability, with respect to either square integrable, or multiply-periodic, or just co-periodic perturbations. Indeed, under the assumption madeearlier in (8)-(9)-(10) that H = H ( v, u, v x ) with ∂ H ∂v x = κ ( v ) > ∇ u H = T ( v ) > , the eigenvalue equation AU = τ U is equivalent to a system of ( N + 3) ODEs(because it involves three derivatives of v ). If F ( · ; τ ) denotes its fundamentalsolution, the existence of a nontrivial U such that AU = τ U , U ( · + Ξ) = e iν U , is equivalent to D ( τ, ν ) = 0, where D ( τ, ν ) := det( F (Ξ; τ ) − e iν ) . This D = D ( τ, ν ) has been called an Evans function. According to its definition, D ( τ,
0) = 0 means that τ is an eigenvalue of A on L ( R / Ξ Z ). In other words, if D ( · ; 0) vanishes somewhere outside the imaginary axis, the wave U is unstablewith respect to co-periodic perturbations. Similarly, if for any n ∈ N ∗ there isa zero of D ( · , π/n ) outside the imaginary axis, the wave U is unstable withrespect to perturbations of period n Ξ. If for any ν ∈ R / π Z , D ( · , ν ) has a zero10utside the imaginary axis, then this zero is an eigenvalue of A on L ∞ ( R ), andalso belongs to the spectrum of A on L ( R ), σ ( A ) = [ ν ∈ R / π Z σ ( A ν ) , which means that the wave U is unstable with respect to both bounded andsquare integrable perturbations.Therefore, locating the zeroes of D ( · , ν ) when ν varies over R / π Z providesvaluable information on the stability of the wave U . The ‘only’ problem with D is that it is not known explicitly in general. If we are not to rely on numericalcomputations, we can only determine some of its asymptotic behaviors. Thisis often sufficient to prove instability results. A most elementary way concerns co-periodic instability . Indeed, since the operator A is real-valued, the function D ( · ,
0) can be constructed so as to be real-valued too. In this case, finding a zeroof D ( · ,
0) on (0 , + ∞ ) may just be a matter of applying the mean value theorem,once we know the behavior of D ( τ,
0) for | τ | ≪ τ ≫ τ ∈ R . Anotherpossibility is to detect side-band instability , which occurs when a zero of D ( · , ν )bifurcates from 0 into the right half-plane for | ν | ≪ J isnot onto. This is a recurrent difficulty with KdV-like PDEs. We consider an open set Ω of ( µ, λ , c ) and assume that we have a smoothmapping ( µ, λ , c ) ∈ Ω ( U , Ξ) such that (6)-(7) hold true with U (0) = U (Ξ),and v x (0) = v x (Ξ) = 0. We are interested in mild modulations of the wave U = U ( x − ct ), in which ( µ, λ , c ) will vary according to a slow time T = εt andon a small length X = εx , with ε ≪
1. The so-called modulated equations willconsist of conservation laws in the (
X, T ) variables for • the wave number, k = 1 / Ξ, • the mean value of the wave, M := k R Ξ0 U d x , • the mean value of the impulse, P := k R Ξ0 Q ( U ) d x .Before writing down these equations, let us see whether the mapping ( µ, λ , c ) ( k, M , P ) has any chance to be a diffeomorphism. A ‘natural’ condition for thisto occur turns out to depend on the Hessian ofΘ( µ, λ , c ) := Z Ξ0 ( H ( U , v x ) + c Q ( U ) + λ · U + µ ) d x , (12)11s a function of its ( N +2) variables. This is because Θ( µ, λ , c ) coincides with the action of the profile ODEs (6), when viewed as a Hamiltonian system associatedwith the Hamiltonian L L . Indeed, by (7), we have v x ∂ H ∂v x ( U , v x ) − H [ U ] − c Q ( U ) − λ · U = µ , hence by change of variableΘ( µ, λ , c ) = I ∂ H ∂v x ( U , v x ) d v , where the symbol H stands for the integral in the ( v, v x )-plane along the orbitdescribed by v . Proposition 1.
Assume that H = H ( U , v x ) is smooth, and that we have asmooth mapping ( µ, λ , c ) ∈ Ω ( U , Ξ) s.t. (6) - (7) hold true, and U (0) = U (Ξ) , v x (0) = v x (Ξ) = 0 . Then the function Θ defined in (12) is also smooth, and we have ∂ Θ ∂µ = Ξ , ∂ Θ ∂c = Z Ξ0 Q ( U ) d x , ∇ λ Θ = Z Ξ0 U d x . (13) Proof.
This is a calculus exercise. Denoting for simplicity by m the function m ( U , v x ; c, λ , µ ) := L ( U , v x ; c, λ ) + µ = H ( U , v x ) + c Q ( U ) + λ · U + µ , if a is any of the parameters c, λ α , µ , and if we denote by a subscript derivationwith respect to a , we haveΘ a = Z Ξ0 m a ( U , v x ; c, λ , µ ) d x + Ξ a m ( U (Ξ) , v x (Ξ); c, λ , µ )+ Z Ξ0 (cid:16) U a · ∇ U m ( U , v x ; c, λ , µ ) + v x,a ∂ H ∂v x ( U , v x ) (cid:17)(cid:17) d x. The announced formulas rely on the observation that all but the first term inthe right-hand side here above equal zero. To show this, let us insist on the factthat, by (7), m ( U , v x ; c, λ , µ ) = v x ∂ H ∂v x ( U , v x ) . Since v x (Ξ) = 0, we thus readily see that m ( U (Ξ) , v x (Ξ); c, λ , µ ) = 0. In orderto deal with the last, integral term in Θ a , we observe that v x,a = ∂ x v a , andmake an integration by parts, in which the boundary terms cancel out, againbecause v x (0) = v x (Ξ) = 0. This yields Z Ξ0 (cid:16) U a · ∇ U m ( U , v x ; c, λ , µ ) + v x,a ∂ H ∂v x ( U , v x ) (cid:17)(cid:17) d x = Z Ξ0 U a · E L [ U ] d x , which is equal to zero because of (6). 12 orollary 1. Under the assumptions of Proposition 1, the mapping ( µ, λ , c ) ∈ Ω ( k, M , P ) is a diffeomorphism if and only if it is one-to-one and det (cid:16) Hess Θ( µ, λ , c ) (cid:17) = 0 , ∀ ( µ, λ , c ) ∈ Ω . Proof.
The mapping ( µ, λ , c ) ∈ Ω ( k = 1 / Ξ , M = k R Ξ0 U d x, P = k R Ξ0 Q ( U )d x )is clearly a diffeomorphism if and only if( µ, λ , c ) ∈ Ω (Ξ , R Ξ0 U d x, R Ξ0 Q ( U ) d x )is so. By Proposition 1, we have that(Ξ , R Ξ0 U d x , . . . , R Ξ0 U N d x , R Ξ0 Q ( U ) d x ) T = ∇ Θ( µ, λ , c ) , and the Jacobian matrix of ( µ, λ , c )
7→ ∇ Θ( µ, λ , c ) is by definition the Hessianof Θ.Let us assume that ( µ, λ , c ) ∈ Ω ( k, M , P ) is indeed a diffeomorphism.Then periodic wave profiles may be parametrized by ( k, M , P ) instead of ( µ, λ , c ).In what follows, we make the dependence on ( k, M , P ) explicit by denoting suchprofiles by U ( k, M ,P ) , which in addition we rescale so that they all have the sameperiod, say one. Then each of them is associated with a travelling wave solutionto (1) by setting U ( t, x ) = U ( k, M ,P ) ( kx + ω ( k, M , P ) t ) , of speed c = c ( k, M , P ), and time frequency ω = ω ( k, M , P ) := − k c ( k, M , P ).We are interested in solutions to (1) taking the form of slowly modulatedwave trains U ( t, x ) = U ( k, M ,P )( εt,εx ) (cid:0) ε φ ( εt, εx ) (cid:1) + O ( ε ) , with φ = φ ( T, X ) such that φ X = k and φ T = ω . (Note that when ( k, M , P ) isindependent of ( T, X ), we just recover exact, periodic travelling wave solutions.)Whitham’s averaged equations consist of conservation laws for ( k, M , P ) =( k, M , P )( T, X ) obtained by formal asymptotic expansions. In fact, the equa-tion on k is just obtained by the Schwarz lemma applied to the phase φ , ∂ T k + ∂ X ( ck ) = 0 . (14)The equations on M and P are derived by plugging the more precise ansatz U ( t, x ) = U ( εt, εx, φ ( εt, εx ) /ε ) + ε U ( εt, εx, φ ( εt, εx ) /ε, ε ) + o ( ε ) , in (1) and (3) respectively, assuming that U and U are 1-periodic in theirthird variable θ (the rescaled phase). The O (1) terms vanish provided that U ( T, X, θ ) = U ( k, M ,P )( T,X ) ( θ ) . O ( ε ) terms involving U cancel out when averaging, andwe receive the equations ∂ T M = B ∂ X h E H k [ U ( k, M ,P ) ] i , (15) ∂ T P = ∂ X h U · E H k [ U ( k, M ,P ) ] + L H k [ U ( k, M ,P ) ] i . (16)Here above, we have used the shortcut H k := H ( U , k U θ ), and the Euler oper-ator E and Legendre transform L act as operators on functions of the rescaledvariable θ . Of course we may simplify and write h E H k [ U ] i = h∇ U H k ( U , k U θ ) i in (15). However, this is not as nice a simplification as the reformulation of theaveraged equations given below. Proposition 2.
Under the assumptions of Proposition 1, the system of equa-tions in (14) - (15) - (16) equivalently reads, as far as smooth solutions are con-cerned, ∂ T (cid:16) ∂ Θ ∂µ (cid:17) + c ∂ X (cid:16) ∂ Θ ∂µ (cid:17) − (cid:16) ∂ Θ ∂µ (cid:17) ∂ X c = 0 ,∂ T (cid:0) ∇ λ Θ (cid:1) + c ∂ X (cid:0) ∇ λ Θ (cid:1) + (cid:16) ∂ Θ ∂µ (cid:17) B ∂ X λ = 0 ,∂ T (cid:16) ∂ Θ ∂c (cid:17) + c ∂ X (cid:16) ∂ Θ ∂c (cid:17) − (cid:16) ∂ Θ ∂µ (cid:17) ∂ X µ = 0 . (17) or in quasilinear form, Σ ∂ T W + ( c Σ + Θ µ S ) ∂ X W = 0 (18) with W T := ( µ, λ T , c ) , Σ := Hess Θ , Θ µ = ∂ Θ ∂µ (at constant λ , c ), S := · · · −
10 0 ... B ... − · · · . Proof.
Recalling that ∂ Θ ∂µ = Ξ = 1 /k , and multiplying Eq. (14) by − Ξ , we readily obtain the first equation in (17).The other ones require a little more manipulations. Regarding (15), we use that M = k ∇ λ Θ , that by the profile equation (6) (after rescaling), E H k [ U ( k, M ,P ) ] = − c B − U ( k, M ,P ) − λ , B h E H k [ U ( k, M ,P ) ] i = − c M − B λ , and we eliminate the factor k by using again (14). We proceed in a similarmanner for (16), using that P = k (cid:16) ∂ Θ ∂c (cid:17) , and that by the profile equations in (6)-(7), U · E H k [ U ( k, M ,P ) ] + L H k [ U ( k, M ,P ) ] = µ − c Q [ U ( k, M ,P ) ] . Remark 1.
Would Σ = Hess Θ be positive definite, (17) would automaticallybelong to the class of symmetrizable hyperbolic systems, in view of its quasi-linear form of (18) . However, as we shall see in Section 3 (Theorem 1), thedefiniteness of Hess Θ is often incompatible with co-periodic stability. In otherwords, despite the nice, ‘symmetric’ form of the modulated equations (17) , theirwell-posedness is far from being automatic, especially in case of co-periodic sta-bility. The simultaneous occurrence of modulational stability and co-periodicstability remains possible though. This is in contrast with the framework of‘quasi-gradient systems’ considered in [18], for which it has been shown thatco-periodic and modulational stability are indeed incompatible. Theorem 1.
Under the structural conditions in (8) - (9) - (10) , and the assump-tions of Proposition 1, • for N = 1 , if det( Hess Θ) > then the wave is spectrally unstable withrespect to co-periodic perturbations; • for N = 2 , if det( Hess Θ) < then the wave is spectrally unstable withrespect to co-periodic perturbations. The first point is a slight generalization - with variable κ ( v ) - of what wasshown by Bronski and Johnson [10]. The second point has been shown in [6] bymeans of an Evans function computation.In both cases, the detected instability corresponds to a real positive unstableeigenvalue. Indeed, the sign criteria here above stem from a mod 2 count of sucheigenvalues. 15 .2 Modulational instability implies side-band instability A necessary condition for spectral stability is modulational stability. This wasshown by Serre [19], and by Oh and Zumbrun [16] for viscous periodic waves.In our framework, we have the following
Theorem 2.
Assume that U is a periodic travelling wave profile, that theset of nearby profiles is, up to translations, an ( N + 2) -dimensional manifoldparametrized by ( µ, c, λ ) , and that the generalized kernel of A in the space of Ξ -periodic functions is of dimension N + 2 . Then the system of modulated equa-tions in (14) - (15) - (16) , or equivalently (17) , is indeed an evolution system (inother words, Hess Θ is nonsingular), and if it admits a nonreal characteristicspeed then for any small enough Floquet exponent ν , the operator A ν admits a(small) unstable eigenvalue. This is a concatenation of results shown in [5].If Ξ = Θ µ = 0 (which we have implicitly assumed up to now), the hyperbol-icity of (18) is equivalent, by change of frame and rescaling, to that of Σ ∂ T W + S ∂ X W = 0 . Assuming that Σ = Hess
Θ is nonsingular and noting that S is always non-singular (because we have assumed that B is so), we thus see that the localwell-posedness of the averaged equations in (17) is equivalent to the fact that S − Σ is diagonalizable on R . Theorem 2 here above shows that spectral stabilityimplies at least that the eigenvalues of S − Σ are real. Case N = 1 (KdV). We have S − = S ∈ R × , and S − Σ = − Θ cµ − Θ cλ − Θ cc Θ λµ Θ λλ Θ λc − Θ µµ − Θ µλ − Θ µc . Then a necessary criterion for spectral stability is that the discriminant of thecharacteristic polynomial of this matrix be nonnegative. This criterion dependsonly on the second-order derivatives of the action Θ.For the case N = 2 (EK), a 4 × Small-amplitude limit.
A necessary condition for modulational stability ofsmall-amplitude waves is two-fold and requires: 1) the hyperbolicity of thereduced system obtained in the zero-dispersion limit; 2) the so-called Benjamin–Feir–Lighthill criterion. We refer to [5] for more details. Observe in particularthat the first condition is trivial in the case N = 1 (because all scalar, first orderconservation laws are hyperbolic), and has hardly ever been noticed. In the case N = 2, and in particular for the Euler–Korteweg system, it requires that theEuler system be hyperbolic at the mean value of the wave. This is a nontrivial16ondition, which rules out some of the periodic waves in the Euler–Kortewegsystem when it is endowed with, for instance, the van der Waals pressure law.As to the Benjamin–Feir–Lighthill criterion, it is famous for characterizing theunstable Stokes waves. We assume as before that H = H ( v, u, v x ), and use the short notation H s for H s ( R / Ξ Z ) × ( L ( R / Ξ Z )) N − . Observe in particular that the functional F ( c, λ ,µ ) : U Z Ξ0 ( H ( U , U x ) + c Q ( U ) + λ · U + µ ) d x is well-defined on H . What we call a Grillakis–Shatah–Strauss (GSS) criterionis a set of inequalities regarding the second derivatives of Θ ensuring that thefunctional F ( c, λ ,µ ) admits a local minimum at U (and any one of its translates)on H ∩ C with C = { U ∈ H ; R Ξ0 Q ( U ) d x = R Ξ0 Q ( U ) d x , R Ξ0 U d x = R Ξ0 U d x } . By a Taylor expansion argument, seeking a GSS criterion amounts to findingconditions under which the operator A = Hess ( H + c Q )[ U ] is nonnegative on H ∩ T U C with T U C := { U ∈ H ; R Ξ0 U · ∇ U Q ( U ) d x = 0 , R Ξ0 U d x = 0 } . Even though it might not be clear at once that such criteria exist, they do. As wehave recalled above (in § M cc >
0, where M is to solitary waves what our Θ is to periodic waves.Behind this criterion is a rather general result, pointed out at various places andshown in most generality by Pogan, Scheel, and Zumbrun [18], which makes theconnection between the negative signatures of the unconstrained version of theHessian of the functional we are trying to minimize, of its constrained version,and of the Jacobian matrix of the values of the constraints in terms of theLagrange multipliers. More explicitly, in our framework with our notations, andunder some ‘generic’ assumptions, the negative signature n ( A ) of the operator A is found to be equal to the negative signature n ( A | T U C ) of its restriction to T U C plus the negative signature n ( − C ) of − C , where C is the Jacobian matrixof the values of the constraints, R Ξ0 U , R Ξ0 Q ( U ), in terms of the Lagrangemultipliers ( λ , c ) when the period Ξ is fixed. The counterpart of this matrix C for solitary waves is just the scalar M cc , in which case we readily see that M cc > n ( − C ) = 1. We now give a version of the Pogan–Scheel–Zumbrun theorem adapted to our framework and notations for periodicwaves. 17 heorem 3. Under the hypotheses of Proposition 1, we assume moreover that Ξ µ = 0 , and that C := ˇ ∇ Θ − ˇ ∇ Ξ ⊗ ˇ ∇ ΞΞ µ takes nonsingular values, with Θ defined as in (12) by Θ( µ, λ , c ) := Z Ξ0 ( H ( U , U x ) + c Q ( U ) + λ · U + µ ) d x , and ˇ ∇ being a shortcut for the gradient with respect to ( λ , c ) at fixed µ . Then,denoting A := Hess ( H + c Q )[ U ] , we have n ( A ) = n ( A | T U C ) + n ( − C ) . Proof.
By assumption, the period Ξ of a given profile U is a smooth functionof the N + 2 parameters ( µ, λ , c ). The fact that Ξ µ = 0 implies by the implicitfunction theorem that µ can be viewed as a smooth function µ = µ (Ξ , λ , c ), andthat ∂µ∂λ α = − Ξ λ α Ξ µ , ∂µ∂c = − Ξ c Ξ µ . (19)Since we are interested in the signature of A on H = ( L ( R / Ξ Z )) N , we shallmostly concentrate on travelling profiles of fixed period Ξ, which are solution to(6)-(7) with µ = µ (Ξ , λ , c ). For such profiles, let us denote by q the constraintsmapping q : ( λ , c ) ( R Ξ0 U d x, R Ξ0 Q ( U ) d x ) , and q α its components, α ∈ { , . . . , N + 1 } , q α ( λ , c ) := Z Ξ0 U α d x , α ∈ { , . . . , N } , q N +1 ( λ , c ) := Z Ξ0 Q ( U ) d x . From Eqs (13) in Proposition 1 and Eqs in (19), we infer that for α , β ≤ N , ∂q β ∂λ α = Θ λ α λ β − Ξ λ α Ξ λ β Ξ µ , ∂q β ∂c = Θ cλ β − Ξ c Ξ λ β Ξ µ ,∂q N +1 ∂λ α = Θ λ α c − Ξ λ α Ξ c Ξ µ , ∂q N +1 ∂c = Θ cc − Ξ c Ξ c Ξ µ . In other words, the Jacobian matrix of q is indeed C = ˇ ∇ Θ − ˇ ∇ Ξ ⊗ ˇ ∇ ΞΞ µ = ˇ ∇ Θ − ˇ ∇ Θ µ ⊗ ˇ ∇ Θ µ Θ µµ . Now, by differentiating (6) with respect to µ , λ or c , we see that A (cid:16) U λ α − Ξ λα Ξ µ U µ (cid:17) = − e α , A (cid:16) U c − Ξ c Ξ µ U µ (cid:17) = − ∇ U Q ( U ) , (20)18here e α denotes the α -th vector of the ‘canonical’ basis of R N , hence thealternative expression for α , β ≤ N , C α,β = − D(cid:16) U λ β − Ξ λβ Ξ µ U µ (cid:17) · A (cid:16) U λ α − Ξ λα Ξ µ U µ (cid:17)E C α,N +1 = − D(cid:16) U c − Ξ c Ξ µ U µ (cid:17) · A (cid:16) U λ α − Ξ λα Ξ µ U µ (cid:17)E C N +1 ,β = − D(cid:16) U λ β − Ξ λβ Ξ µ U µ (cid:17) · A (cid:16) U c − Ξ c Ξ µ U µ (cid:17)E C N +1 ,N +1 = − D(cid:16) U c − Ξ c Ξ µ U µ (cid:17) · A (cid:16) U c − Ξ c Ξ µ U µ (cid:17)E (21)where h · i denotes the inner product in L ( R / Ξ Z ; R N ). This implies, if C isnonsingular, that H = Span( U λ − Ξ λ Ξ µ U µ , . . . , U λ N − Ξ λN Ξ µ U µ , U c − Ξ c Ξ µ U µ ) ⊕ T U C ,T U C = { U ∈ H ; h U · ∇ U Q ( U ) i = 0 , h U i = 0 } . As a matter of fact, Equations in (20)-(21) imply that for any V ∈ H , there isone and only one ( a , . . . , a N +1 , U ) ∈ R N +1 × T U C such that V = a (cid:16) U λ − Ξ λ Ξ µ U µ (cid:17) + · · · + a N (cid:16) U λ N − Ξ λN Ξ µ U µ (cid:17) + a N +1 (cid:16) U c − Ξ c Ξ µ U µ (cid:17) + U , which can be computed by solving the ( N + 1) × ( N + 1) system R Ξ0 V d x... R Ξ0 V N d x R Ξ0 V · ∇ U Q ( U ) d x = C a ...a N a N +1 . In order to conclude, let us denote by Π the orthogonal projection onto thespace Span( U λ − Ξ λ Ξ µ U µ , . . . , U λ N − Ξ λN Ξ µ U µ , U c − Ξ c Ξ µ U µ ) , and by Π the orthogonal projection onto T U C . We readily see that for all U ∈ T U C and V ∈ Span( U λ − Ξ λ Ξ µ U µ , . . . , U λ N − Ξ λN Ξ µ U µ , U c − Ξ c Ξ µ U µ ), h U · A V i = 0 , hence Π A Π = 0 , Π A Π = 0 . Therefore, for all V ∈ D ( A ) = H , h V · A V i = h V · Π A Π V i + h V · Π A Π V i . From this relation we see that the negative signature of A is the sum of thoseof Π A Π and Π A Π . The latter is the negative signature of A | T U C , bydefinition of the projection Π , while the former coincides with the negativesignature of − C by definition of the projection Π and by the expression of C in (21). 19ote that C is a codimension ( N + 1) manifold of H . As a matter of fact,the constraints defining C are ‘full rank’, in the sense that for all ( m , q ) ∈ R N +1 ,there exists U ∈ H such that Z Ξ0 U d x = m , Z Ξ0 U · ∇ U Q ( U ) d x = q . Recalling that ∇ U Q ( U ) = B − U , we may take for instance U = m Ξ + a B − U xx with a = m · B − M − q R Ξ0 k B − U x k d x . Corollary 2.
If the negative signatures of the operator A and of the matrix C defined in Theorem 3 are equal, then the periodic travelling wave ( x, t ) U ( x − ct ) is (conditionally) orbitally stable to co-periodic perturbations.Proof. From Theorem 3 we infer that the negative signature of A | T U C is zero.In other words, the functional F ( c, λ ,µ ) does have a local minimum at U on H ∩ C . (This follows from a Taylor expansion and the density of D ( A ) in H .)The fact that it is not a strict minimum can be coped with by ‘factoring out’the translation-invariance problem in the usual way. Namely, by the implicitfunction theorem, there exists a tubular neighborhood N in L ( R / Ξ Z ) of U and all its translates U ( · + ξ ) for ξ ∈ R , and a smooth mapping s : N → R suchthat for all U ∈ N , U ( · − s ( U )) − U is orthogonal to U x . As a consequence,up to diminishing N , we can find an α > U ∈ N ∩ H ∩ C , F ( c, λ ,µ ) [ U ] − F ( c, λ ,µ ) [ U ] = Z Ξ0 ( H ( U , U x ) − H ( U , U x )) d x ≥ α k U − U ( · + s ( U )) k H . This enables us to show the following, conditional stability result. If H ⊂ H issuch that the Cauchy problem associated with (1) is locally well-posed in H , ifwe denote by T ( U ) the maximal time of existence of the solution U of (1) in H with initial data U ∈ H , ∀ ε > , ∃ δ > ∀ U ∈ H ; k U − U k H ≤ δ ⇒ ∀ t ∈ [0 , T ( U )) , inf s ∈ R k U ( t, · ) − U ( · + s ) k H ≤ ε . The proof works by contradiction, as in [13, 8], even though an alternative,direct proof as in [14] is also possible. Assume there exist ε >
0, and a sequenceof initial data U ,n ∈ H such that inf s ∈ R k U ,n − U ( · + s ) k H goes to zero whilesup t ∈ [0 ,T ( U )) inf s ∈ R k U n ( t, · ) − U ( · + s ) k H > ε . Without loss of generality, we can assume that the tubular neighborhood of U of radius 2 ε is contained in N . We choose t n to be the least value such thatinf s ∈ R k U n ( t n , · ) − U ( · + s ) k H = ε .
20y invariance of R Ξ0 H [ U ] d x , R Ξ0 Q [ U ] d x , and R Ξ0 U d x , with respect to timeevolution and spatial translations, we have Z Ξ0 H [ U n ( t n )] d x = Z Ξ0 H [ U ,n ] d x → Z Ξ0 H [ U ] d x , Z Ξ0 Q ( U n ( t n )) d x = Z Ξ0 Q ( U ,n ) d x → Z Ξ0 Q ( U ) d x , Z Ξ0 U n ( t n ) d x = Z Ξ0 U ,n d x → Z Ξ0 U d x . This implies, by using the full-rank property mentioned above and the submer-sion theorem that we can pick for all n some V n ∈ N ∩ H ∩ C such that k V n − U n ( t n ) k H goes to zero, as well as Z Ξ0 ( H [ V n ] − H [ U ]) d x → . Therefore, k V n ( · ) − U ( · + s ( V n )) k H ≤ α Z Ξ0 ( H [ V n ] − H [ U ]) d x → , hence k U n ( t n , · ) − U ( · + s ( V n )) k → t n . Remark 2.
For (KdV), a case in which N = 1 , it has been shown by Johnson[14] that a periodic wave is orbitally stable to co-periodic perturbations underthe two conditions Θ µµ > , det( Hess Θ) < . (22) It is not difficult to see that these assumptions imply that the constraints matrix C has signature ( − , +) . Indeed, for N = 1 we have C = 1Θ µµ (cid:18) Θ µµ Θ λλ − Θ µλ Θ λµ Θ µµ Θ cλ − Θ µc Θ λµ Θ µµ Θ λc − Θ µλ Θ cµ Θ µµ Θ cc − Θ µc Θ cµ (cid:19) , and a bit of algebra shows that Θ µµ det C = det( Hess Θ) , so that if (22) hold true, det( − C ) = det C < thus n ( − C ) = 1 . The result thenfollows from [14, Lemma 4.2], which proves that Θ µµ > implies n ( A ) = 1 . ppendix Table of examples N U J H Q (KdV) 1 v ∂ x v x + f ( v ) v (EKL) 2 (cid:18) vw (cid:19) (cid:18) ∂ y ∂ y (cid:19) u + `e ( v, v y ) vw (EKE) 2 (cid:18) ρu (cid:19) − (cid:18) ∂ x ∂ x (cid:19) ρu + E ( ρ, ρ x ) − ρu (B) 2 (cid:18) χχ t (cid:19) (cid:18) − (cid:19) χ t + W ( χ x ) ± χ xx χ t χ x (NLW) 2 (cid:18) χχ t (cid:19) (cid:18) − (cid:19) χ t + χ x + V ( χ ) χ t χ x (NLS) 2 (cid:18) Re ψ Im ψ (cid:19) (cid:18) − (cid:19) | ψ x | + F ( | ψ | ) − Im ( ψ ψ x ) Sturm–Liouville argument
Assume that H = H ( v, u, v x ) = E ( v, v x ) + T ( v, u ) , ∂ E ∂v x =: κ ( v ) > , ∇ u T =: T ( v ) > , Q = Q ( U ) = U · B − U , U T = ( v, u T ) , B − = (cid:18) a b T b N − (cid:19) , with κ ( v ) > T ( v ) symmetric definite positive for all v , and b = 0. Theprofile equations E ( H + c Q )[ U ] + λ = 0 equivalently read ( E E [ v ] + ∂ v T ( v, u ) + c ( a v + b · u ) + λ = 0 , ∇ u T ( v, u ) + c v b + ˇ λ = 0 , and their integrated version L ( H + c Q + λ · U )[ U ] = µ reads L ℓ [ v ] = µ , where ℓ = ℓ ( v, v x ; c, λ ) is defined by ℓ = E ( v, v x ) + T ( v, f ( v ; c, ˇ λ )) + c ( av + v b · f ( v ; c, ˇ λ )) + λ v + ˇ λ · f ( v ; c, ˇ λ ) , ( v ; c, ˇ λ ) := − T ( v ) − ( ∇ u T ( v,
0) + c v b + ˇ λ ) . Defining A := Hess ( H + c Q )( U ) = (cid:18) Hess E [ v ] + ∂ v T ( v, u ) + ca ( ∂ v ∇ u T ( v, u ) + cb ) T ∂ v ∇ u T ( v, u ) + cb T ( v ) (cid:19) , we see by differentiating with respect to x in the profile equations that A U x = 0,or equivalently ( Hess E [ v ] v x + v x ∂ v T ( v, u ) + u x · ∂ v ∇ u T ( v, u ) + c ( av x + b · u x ) = 0 ,v x ∂ v ∇ u T ( v, u ) + T ( v ) u x + c v x b = 0 . This can be shown to imply that a v x = 0 with a := Hess ℓ [ v ]. A simpleralternative to show that a v x = 0 consists in differentiating with respect to x inthe Euler–Lagrange equation E ℓ [ v ] = 0. If in addition E depends quadraticallyon v x , then a is of the form − ∂ x κ ( v ) ∂ x + q ( x ), where q ( x ) depends on the profile v - which depends itself on ( c, λ , µ ) - and on the parameters c, λ . Hence a is a Sturm–Liouville operator with Ξ-periodic coefficients. The fact that a v x = 0and v is Ξ-periodic (and not constant) implies that a has at least one, and atmost two negative eigenvalues (see for instance [20, Theorem 5.37]). References [1] T. B. Benjamin. The stability of solitary waves.
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