Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability
Adam L. Binswanger, Mark A. Hoefer, Boaz Ilan, Patrick Sprenger
WWHITHAM MODULATION THEORY FOR GENERALIZEDWHITHAM EQUATIONS AND A GENERAL CRITERION FORMODULATIONAL INSTABILITY
ADAM L. BINSWANGER , MARK A. HOEFER , BOAZ ILAN , AND PATRICK SPRENGER , Abstract.
The Whitham equation was proposed as a model for surface water wavesthat combines the quadratic flux nonlinearity f ( u ) = u of the Korteweg-de Vriesequation and the full linear dispersion relation Ω( k ) = √ k tanh k of uni-directionalgravity water waves in suitably scaled variables. This paper proposes and analyzes ageneralization of Whitham’s model to unidirectional nonlinear wave equations consist-ing of a general nonlinear flux function f ( u ) and a general linear dispersion relationΩ( k ). Assuming the existence of periodic traveling wave solutions to this generalizedWhitham equation, their slow modulations are studied in the context of Whithammodulation theory. A multiple scales calculation yields the modulation equations, asystem of three conservation laws that describe the slow evolution of the periodic trav-eling wave’s wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit,simple criteria in terms of general f ( u ) and Ω( k ) establishing the strict hyperbolicityand genuine nonlinearity of the modulation equations are determined. This result isinterpreted as a generalized Lighthill-Whitham criterion for modulational instability. Introduction
Scalar dispersive hydrodynamic equations model nonlinear wave motion in systemswhere dissipation is negligible with respect to wave dispersion. Many nonlinear wavemodels are derived using a multiple scale procedure in the small amplitude, long wave-length regime [1]. The canonical model equation that describes the unidirectional prop-agation of weakly nonlinear long waves is the Korteweg-de Vries (KdV) equation u t + (cid:18) u (cid:19) x + u xxx = 0 , (1)which arises universally in systems exhibiting weak quadratic nonlinearity and long-wave, third order dispersion [52]. Equation (1) can be derived, for example, as a modelof the free surface displacement u ( x, t ) of an incompressible fluid with motion in onespatial dimension. In this scenario, the KdV equation (1) provides an accurate de-scription of free surface dynamics for some phenomena–for instance the evolution ofbroad disturbances–but fails to capture short wave phenomena such as sharp profiles,e.g. peaking waves. In a 1967 paper, Whitham proposed an alternative model equationconsisting of the same quadratic nonlinearity as the KdV equation (1) while capturingthe full linear dispersion of waves moving in the positive x -direction in order to modelwave peaking [51]. The so-called Whitham equation for surface for water waves is given Date : September 8, 2020. a r X i v : . [ n li n . PS ] S e p A. L. BINSWANGER, M. A. HOEFER, B. ILAN, AND P. SPRENGER in nondimensional coordinates by u t + (cid:18) u (cid:19) x + K ∗ u x = 0 , (2)where K ∗ u x is defined as a Fourier multiplier with the symbol (cid:92) K ∗ g ( x ) = (cid:115) tanh qq (cid:98) g ( q ) , (3)and (cid:98) g ( q ) is the Fourier transform of g ( x ) defined by the pair g ( x ) = 12 π (cid:90) R (cid:98) g ( q ) e iqx d q, (cid:98) g ( q ) = (cid:90) R g ( x ) e − iqx d x, (4)for g ( x ) ∈ L ( R ). Thus K is a convolution kernel whose Fourier transform is the linearphase velocity for unidirectional water waves (cid:112) tanh q/q . The Whitham equation (3)can be derived via an asymptotic expansion of the Hamiltonian formulation of the Eulerequations with free surface boundary conditions [37, 14]. The existence and numericalcomputation of smooth periodic, traveling wave solutions was established in [17]. Sub-sequently, the existence of limiting periodic solutions with a cusp, i.e. peaked waves,has been established [18]. Periodic solutions were shown to be unstable with respect tolong wavelength perturbations in sufficiently deep water [41]. Solitary wave solutionsare also known to exist [16]. From a modeling perspective, Eq. (3) outperforms theKdV equation in its approximation of the free surface dynamics of a water wave. Thisis shown both in numerical simulation of the Euler equations [37] and in experimentalwave tanks [46, 9]. The success of the Whitham equation (5) as a model for surfacewater waves motivates consideration of full-dispersion models in the context of otherphysical systems. For example, models similar to equation (2) with a modified convolu-tion operator have been used to model water waves underneath an elastic ice sheet [14]and internal waves at a two-fluid interface, one with infinite depth [27].In this manuscript, we propose a generalization of (2) where the quadratic nonlinearterm is replaced by a general flux and the linear dispersion is similarly determined by aconvolution. The generalized Whitham equation takes the form(5) u t + f ( u ) x + K ∗ u x = 0 , where the convolution K ∗ g ( x ) is defined by the Fourier multiplier (cid:92) K ∗ g ( x ) = Ω( q ) q (cid:98) g ( q ) . Here, f ( u ) and Ω( k ) are the problem dependent nonlinear hydrodynamic flux and lineardispersion relation, respectively. The dispersion relation of small amplitude, sinusoidalwaves on a nonzero background mean, u , depend on the nonlinear flux and linear dis-persion according to ω ( u, k ) = f (cid:48) ( u ) k + Ω( k ) . (6) HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 3
It should be noted that a more complex dependence on the wave mean of the lineardispersion, e.g. in models with nonlinear dispersive terms, is not included in this gener-alized Whitham equation (5) (see ref. [29] for a discussion of other full dispersion modelsfor water waves). However, the class of generalized Whitham equations (5) is quite gen-eral. Setting f ( u ) = u and Ω( k ) = − k recovers the KdV equation (1), and otherchoices of nonlinear flux and dispersion recover well known scalar models including themodified KdV, Benjamin-Ono, Kawahara, and Gardner equations for example, in addi-tion to the original Whitham equation (2). Consequently, our analysis of the generalizedWhitham equation (5) applies to all of these models. Moreover, the equation’s structurewith a general nonlinear flux and linear dispersion provides a broadly encompassing andappealing universal model of unidirectional nonlinear waves with full linear dispersion.We derive and analyze the Whitham modulation equations for the full dispersiongeneralized Whitham equation (5). The Whitham modulation equations are a systemof conservation laws that describe the small dispersion limit of nonlinear wavetrains.Equivalently, they describe the slow evolution of a nonlinear, periodic wavetrain’s pa-rameters. Various methods exist to derive the Whitham modulation equations includingaveraged conservation laws [50], averaged Lagrangian [49], averaged Hamiltonian [7], ora multiple scale procedure [35]. For the KdV equation (1), the Whitham equations wereproven to describe the zero dispersion limit [31, 32, 33, 48] for L ( R ) data with theinverse scattering transform.The mathematical structure of the Whitham modulation system provides useful infor-mation about the evolution of a nonlinear periodic wavetrain. A particular applicationwe focus on in this manuscript is the modulational instability (MI) of a periodic wave-train. A history is provided in [55]; we highlight some key discoveries. Benjamin andFeir demonstrated the breakup of nonlinear surface water waves propagating over a rel-atively deep, flat bottom [6, 4]. Simultaneously, Whitham developed his modulationtheory and noted that the modulation equations may become elliptic [52]. Subsequentwork by Lighthill [34] made the connection between ellipticity of the modulation equa-tions and MI for weakly nonlinear dispersive waves, where the stability of the nonlinearwavetrain relies on the weakly nonlinear correction, ω , to the linear dispersion relationand the linear dispersion curvature Ω (cid:48)(cid:48) ( k ). This Lighthill-Whitham criterion ω Ω (cid:48)(cid:48) ( k ) < , (7)for modulationally unstable waves applies directly when there is no induced mean flow.The introduction of wave-mean coupling requires a modification of the analysis in [52]. Inboth cases, the coefficient ω and its wave-mean coupled generalization are problem de-pendent. One significant result of the present work is the generalized Lighthill-Whithamcriterion: n ( u, k )Ω (cid:48)(cid:48) ( k ) < f ( u ) and Ω( k ). Its negationdetermines the hyperbolicity, hence the modulational stability of weakly nonlinear peri-odic waves. More recent theoretical developments have proven that weak hyperbolicityof the modulation equations is a necessary condition for the modulational stability ofnonlinear periodic wavetrains [8].This manuscript is organized as follows. In Sec. 2, we summarize our main results:the Whitham modulation equations for the generalized Whitham equation (5) and ageneralized criterion for MI. In Sec. 3, we carry out the derivation of the modulation A. L. BINSWANGER, M. A. HOEFER, B. ILAN, AND P. SPRENGER equations. The modulation equations consist of averages that can be interpreted in theFourier domain via Plancherel’s theorem. The modulation equations in the small am-plitude limit are the later focus of Sec. 3, where we demonstrate that the system ofmodulation equations in the weakly nonlinear regime can be written as an explicit sys-tem that gives the approximate evolution of the modulation variables. In this regime, weidentify properties of the quasilinear system and classify its hyperbolicity/ellipticity andgenuine nonlinearity. The hyperbolicity/ellipticity of the weakly nonlinear modulationequations yields an index that specifies the modulational instability of a finite ampli-tude periodic wave. Modulational instability results are then related to those obtainedfrom the Nonlinear Schr¨odinger (NLS) approximation for the modulated waves yieldingadditional information such as the maximum growth rate and wavenumber associatedwith the instability. In Sec. 4, we apply the MI index to various physical systems thatcan be modeled by Eq. (5). Finally, in Sec. 5 we conclude the manuscript and discussfuture problems related to the present work.2.
Main results
We consider the generalized Whitham equation (5) with the following assumptions
Assumption 1. Ω( k ) is a smooth, real valued, odd function and Ω (cid:48) (0) = 0 . Assumption 2. f ( u ) is smooth and f (cid:48) (0) = 0 . Assumption 3.
There exists a three parameter family of π periodic traveling wavesolutions ϕ ( θ ; p ) , θ = kx − ωt with wavenumber k ( p ) and frequency ω ( p ) to Eq. (5) with p ∈ U ⊂ R . Assumptions 1–3 are satisfied by many models including the (m)KdV, generalizedKdV, Gardner, Kawahara, Benjamin-Ono, Benjamin, and Whitham equations to namea few. We note that by Assumption 1, the linear dispersion relations Ω( k ) we considerexclude those with small wavenumber asymptotics of the form Ω( k ) ∼ k α , α <
1. Thespatial coordinate may be chosen so that parts of assumptions 1 and 2 hold without lossof generality. If f (cid:48) (0) = a and Ω (cid:48) (0) = b with a, b ∈ R , transforming to the coordinateframe (cid:101) x = x − ( a + b ) t , (cid:101) t = t removes the effect of these linear terms.A challenge to derive the Whitham modulation equations for the generalized Whithamequation (5) using standard approaches is the analysis of the nonlocal, convolution term.We found it helpful to conduct a multiple scale analysis in the Fourier domain. Weassume a slowly modulated periodic, traveling wave solution of Eq. (5) of the form u ( x, t ) = ϕ ( θ, X, T ) + (cid:15)u ( θ, X, T ) + . . . , < (cid:15) (cid:28) X = (cid:15)x , T = (cid:15)t are long spatial and temporal scales, respectively and θ is thefast phase defined by θ = θ ( X, T ) θ t = − ω ( X, T ) ,θ x = k ( X, T ) , (9)where ω ( X, T ) is the slowly evolving wave frequency and k ( X, T ) is the slowly modulatedwavenumber. We impose the periodicity requirement ϕ ( θ, X, T ) = ϕ ( θ + 2 π, X, T ), HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 5 u j ( θ, X, T ) = u j ( θ + 2 π, X, T ) for j = 1 , , . . . , and θ, X, T ∈ R so that we may expressthe leading order solution in terms of its Fourier series in θ (10) ϕ ( θ, X, T ) = ∞ (cid:88) n = −∞ ϕ n ( X, T ) e inθ ,where the slowly varying Fourier coefficients ϕ n ( X, T ) admit the Fourier transform pairin the slow spatial scale(11) ϕ n ( X, T ) = 12 π (cid:90) ∞−∞ (cid:98) ϕ n ( Q, T ) e iQX d Q , (cid:98) ϕ n ( Q, T ) = (cid:90) ∞−∞ ϕ n ( X, T ) e − iQX d X .In so doing, we identify the multiscale structure of the spatial wavenumber in the Fourierdomain as the two-scale Fourier representation for ϕϕ ( θ, X, T ) = 12 π ∞ (cid:88) n = −∞ (cid:90) R (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q, (cid:98) ϕ n ( Q, T ) = 12 π (cid:90) π − π (cid:90) R ϕ ( θ, X, T ) e − i ( nθ + QX ) d X d θ, (12)where integers n correspond to harmonics of the rapidly varying, locally periodic waveand Q is the modulation wavenumber. Comparing the two scale Fourier representation(12) with the definitions (4), we can identify the multiscale expansion of the spatialwavenumber in the Fourier domain for this slowly modulated periodic wave to be(13) q = kn + (cid:15)Q, i.e., small deviations in wavenumber from each harmonic. The multiscale expansion ofthe convolution operator is described in the following lemma. Lemma 1.
Assuming a modulated periodic traveling wave of the form (8) with Fourierseries representation (10) , the multiscale expansion of the convolution operator has theform (14)
K ∗ u x ∼ k K ∗ ϕ θ + (cid:15) ( K (cid:48) ∗ ϕ X + k K ∗ u ,θ ) , where the n th Fourier series coefficients are the phase velocity K n = Ω( kn ) kn and groupvelocity K (cid:48) n = Ω (cid:48) ( kn ) evaluated at the n th harmonic. The proof of Lemma 1 can be found in Appendix A. We remark that this result hasthe intuitively appealing interpretation that the two-scale expansion of the convolutionappearing in the generalized Whitham equation (5) in the Fourier domain acts upon thelocally periodic wave in θ by multiplication of the linear phase velocity while acting uponthe modulation in X via multiplication by the group velocity. Utilizing Lemma 1 andPlancherel’s theorem, we derive the Whitham modulation equations for the generalizedWhitham equation (5). A. L. BINSWANGER, M. A. HOEFER, B. ILAN, AND P. SPRENGER
Proposition 1.
Consider the generalized Whitham equation (5) satisfying Assumptions1-3. The Whitham modulation equations for modulated wave solutions ϕ ( θ ; p ( X, T )) are ( ϕ ) T + (cid:16) f ( ϕ ) (cid:17) X = 0(15a) (cid:18) ϕ (cid:19) T + (cid:18) F ( ϕ ) + 12 ϕ K (cid:48) ∗ ϕ (cid:19) X = 0 , (15b) k T + ω X = 0 . (15c) The averaging operator G [ ϕ ]( X, T ) is defined by G [ ϕ ]( X, T ) = 12 π (cid:90) π − π G [ ϕ ( θ, X, T )] d θ, (16) where G is a local function of ϕ and its derivatives, F ( ϕ ) = (cid:90) ϕ sf (cid:48) ( s ) d s, (17) and ϕ K (cid:48) ∗ ϕ = (cid:88) n Ω (cid:48) ( nk ) | ϕ n ( X, T ) | . (18)An oft used set of physical modulation variables are p = [ u, a , k ] T where ¯ u is thewave mean, a is the waveheight, and k is the spatial wavenumber. The parameters arerelated to the periodic wave according to¯ u = 12 π (cid:90) π ϕ ( θ ) dθ, (19a) a = max θ ∈ [0 , π ) ϕ ( θ ) − min θ ∈ [0 , π ) ϕ ( θ ) , (19b) k = 2 πL , (19c)where L is the spatial period of the periodic traveling wave. Two corollaries to Propo-sition 1 in the weakly nonlinear regime are the following. Corollary 1.
The Whitham modulation equations (15) for < a (cid:28) are the conser-vation laws u T + (cid:18) f ( u ) + a f (cid:48)(cid:48) ( u ) (cid:19) X = 0 , (20a) (cid:18) u + a
16 + a A (cid:19) T + ( M ) X = 0 , (20b) k T + (cid:0) f (cid:48) ( u ) k + Ω( k ) + a ω (cid:1) X = 0 , (20c) HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 7 where M = F ( u ) + (cid:18) a
16 + a A (cid:19) ( f (cid:48) ( u ) + uf (cid:48)(cid:48) ( u ))+ a
16 Ω (cid:48) ( k ) + a A
64 Ω (cid:48) (2 k ) , (21) and A = kf (cid:48)(cid:48) ( u )4Ω( k ) − k ) , (22a) ω = k (cid:32) k ( f (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) + 12 f (cid:48)(cid:48)(cid:48) ( u ) (cid:33) . (22b) Corollary 2.
The Whitham modulation equations outlined in Corollary 1 are strictlyhyperbolic if Ω (cid:48) ( k ) (cid:54) = 0 and n ( u, k )Ω (cid:48)(cid:48) ( k ) < , (23) where n ( u, k ) = − k (cid:18) ( f (cid:48)(cid:48) ( u )) Ω (cid:48) ( k ) + k ( f (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) + 12 f (cid:48)(cid:48)(cid:48) ( u ) (cid:19) = − k ( f (cid:48)(cid:48) ( u )) Ω (cid:48) ( k ) − ω (24) Moreover, the strictly hyperbolic Whitham modulation equations (20) are genuinely non-linear if f (cid:48)(cid:48) ( u ) (cid:54) = 0 . Corollary 2 is one of our primary results and can be interpreted as a generalization ofthe aforementioned Lighthill-Whitham criterion (7) to general nonlinear flux and lineardispersion with wave-mean coupling.In the weakly nonlinear regime, the modulations of a carrier wave can also be describedby the Nonlinear Schr¨odinger equation. The NLS approximation provides additionalinformation about the growth of unstable perturbations. These are summarized in thefollowing.
Corollary 3.
The slowly varying, complex envelope, A , of periodic solutions in theweakly nonlinear limit is governed by the NLS equation iA τ + n ( u, k ) | A | A + Ω (cid:48)(cid:48) ( k )2 A ξξ = 0 ,τ = (cid:15) t, ξ = (cid:15) ( x − ω (cid:48) ( k ) t ) , (25) where n ( u, k ) is defined in (24) and (cid:15) = a . Perturbations to the wave envelope of theform A = e inτ (cid:0) αe iκξ + γτ (cid:1) , | α | (cid:28) grow exponentially if n ( u, k )Ω (cid:48)(cid:48) ( k ) > . The maximal growth rate occurs when κ = κ max κ = n ( u, k )Ω (cid:48)(cid:48) ( k ) , A. L. BINSWANGER, M. A. HOEFER, B. ILAN, AND P. SPRENGER and the corresponding maximal growth rate is γ max = | n ( u, k ) | . (27)Properties of the NLS approximation (25) are discussed in Sec. 4.1 and details of thederivation are given in Appendix C.3. Proof of Proposition 1 and Corollaries 1 and 2
Recall that the 2 π -periodic traveling waves ϕ ( θ ) are assumed to be characterized bythree parameters: mean ¯ u , amplitude a , and wavenumber k , as defined previously in Eq.(19). Figure 1 is a sketch of a nonlinear periodic wave with these parameters identified. ¯ u π / k a ' π − π − π π ✓ Figure 1.
Sketch of a nonlinear periodic wave with the physical param-eters: wave mean u , wave amplitude a , and wavenumber k .The Whitham modulation equations (15) are derived using a multiple scale procedure[35]. We seek a modulated 2 π -periodic traveling wave solution of the form of Eq. (8)with phase satisfying Eq. (9). Inserting the multiple scale expansion into Eq. (5), wecollect and study problems appearing at each increasing order in (cid:15) . The leading orderand first order equations are respectively O (1) : − ωϕ θ + kf (cid:48) ( ϕ ) ϕ θ + K ϕ θ = 0(28) O ( (cid:15) ) : L u = − ϕ T − f (cid:48) ( ϕ ) ϕ X − ( K (cid:48) ϕ ) X (29)where the linear operator L is L g = − ωg θ + k ( f (cid:48) ( ϕ ) + f (cid:48)(cid:48) ( ϕ ) ϕ θ ) g θ + k K ∗ g θ . The leading order problem, Eq. (28), is the profile equation for the periodic travelingwave solution, ϕ , of Eq. (5) that was assumed to exist.The Whitham modulation equations are obtained by requiring a 2 π periodic solutionto Eq. (29) to exist in L ([0 , π ]). Equation (29) is solvable if the right hand side isorthogonal to the kernel of the adjoint operator L † (30) L † g = ωg θ − kf (cid:48) ( ϕ ) g θ − K g θ . The proof relies on integration by parts and application of the Plancherel theorem forthe L ([0 , π ]) inner product (cid:104) f, g (cid:105) of real functions v and w (cid:104) v, w (cid:105) = (cid:90) π v ( θ ) w ( θ ) d θ. HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 9
Then (cid:104)K ∗ v θ , w (cid:105) = 2 π (cid:80) n i Ω( kn ) v n w ∗ n so that (cid:104)L v, w (cid:105) = (cid:90) π ( − ωv θ w + k ( f (cid:48) ( ϕ ) v ) θ w + k ( K ∗ v θ ) w ) d θ = ω (cid:90) π vw θ d θ − k (cid:90) π vf (cid:48) ( ϕ ) w θ d θ + 2 πk (cid:88) n i Ω( nk ) v n w ∗ n = (cid:104) v, ωw θ (cid:105) + (cid:104) v, − kf (cid:48) ( ϕ ) w θ (cid:105) − πk (cid:88) n v n ( i Ω( nk ) w n ) ∗ = (cid:104) v, ωw θ (cid:105) + (cid:104) v, − kf (cid:48) ( ϕ ) w θ (cid:105) + (cid:104) v, − k K ∗ w θ (cid:105) = (cid:104) v, L † w (cid:105) ,where · ∗ denotes complex conjugation.By inspection, the kernel of L † (30) includes the linearly independent 2 π -periodicsolutions ϕ and 1. Therefore, two necessary conditions for the existence of 2 π -periodicsolutions to Eq. (29) are (cid:104) , ϕ T + f (cid:48) ( ϕ ) ϕ X + ( K (cid:48) ϕ ) X (cid:105) = 0 , (31) (cid:104) ϕ, ϕ T + f (cid:48) ( ϕ ) ϕ X + ( K (cid:48) ϕ ) X (cid:105) = 0 , (32)yielding the first two Whitham modulation equations. Averages involving the nonlocaloperator can be computed directly by use of the Plancherel theorem. Adding the phasecompatibility condition θ XT = θ T X , we arrive at the Whitham modulation equations(15).The general approach here results in the Whitham modulation equations for any scalarequation that can be written as a generalized Whitham equation (5). Typically, theWhitham modulation equations are derived independently for each model equation. Theapproach presented here yields the Whitham modulation equations for an entire classof equations. The modulation equations (15) exactly recover the modulation equationsfor canonical models such as the KdV equation with appropriate choices of f and Ω.We note that although we chose to use the method of multiple scales to derive theWhitham modulation equations (15), the same results can be obtained using the av-eraged conservation laws approach. The generalized Whitham equation (5) admits thetwo conserved densities for decaying u [38] P = (cid:90) R u d x, (33a) P = (cid:90) R u d x. (33b)With these, we can obtain the first two Whitham modulation equations (15a) and (15b)by computing the averages u t + f ( u ) x + K ∗ u x = 0 , (34a) uu t + uf ( u ) x + u K ∗ u x = 0 , (34b)using the averaging operation defined in Eq. (16) and Lemma 1. Doing so results inexactly Eqs. (15a) and (15b), and the system is completed by adding the conservationof waves Eq. (15c). Weakly nonlinear Whitham modulation equations.
In this section, we utilizea weakly nonlinear, Stokes wave approximation of periodic traveling wave solutions to thegeneralized Whitham equation (5) and express the modulation equations (15) explicitlyin terms of parameters of the modulated periodic wave defined in Eq. (19). The Stokeswave is computed upon seeking a periodic solution for small amplitude 0 < a (cid:28)
1. Astandard perturbation calculation in Appendix B demonstrates that the approximateperiodic traveling wave profile up to O ( a ) is u = u + a θ + a A cos 2 θ + O ( a ) , (35a) ω = f (cid:48) ( u ) k + Ω( k ) + a ω + O ( a )(35b)where A and ω are given in equation (22). The weakly nonlinear modulation equationsare then determined by inserting the expansions (35) into the modulation equations (15).Upon doing so, we obtain the Whitham modulation equations (20). Up to this point, weretained all terms to O ( a ) when the modulation equations are cast in the quasilinearform p T + Bp X = 0 , (36a) B = f (cid:48) ( u ) + f (cid:48)(cid:48)(cid:48) ( u ) a f (cid:48)(cid:48) ( u ) 02 a f (cid:48)(cid:48) ( u ) b a Ω (cid:48)(cid:48) ( k ) f (cid:48)(cid:48) ( u ) k + a ω ,u ω f (cid:48) ( u ) + Ω (cid:48) ( k ) + a ω ,k ,(36b) b = f (cid:48) ( u ) + Ω (cid:48) ( k ) + A − Ω (cid:48) ( k ) + Ω (cid:48) (2 k ) + uf (cid:48)(cid:48) ( u )) a , (36c)where p = [ u, a , k ] T .3.2. Mathematical properties of the weakly nonlinear Whitham modulationequations.
The quasilinear, weakly nonlinear Whitham modulation equations (36) arein a form that is amenable to further analysis, which has implications for the evolution ofweakly nonlinear wavetrains. Notable features of the modulation equations (36) we willidentify here are their hyperbolicity and genuine nonlinearity. Both of these propertiesrely on the eigensystem for the matrix defined in Eq. (36b). Generically, B possessesthree eigenpairs { λ i , v i } i =1 where v i is the right eigenvector in R with correspondingeigenvector λ i . The following definitions are due to Lax [30] (see also Ref. [13]). Thesystem (36a) is called hyperbolic if its 3 eigenvalues are real with linearly independentright eigenvectors. If additionally, the 3 eigenvalues are distinct, then the system is strictly hyperbolic . If all eigenvalues are real but the eigenvectors are linearly dependent,the system is weakly hyperbolic . The characteristic families of a hyperbolic system arethen said to be genuinely nonlinear if(37) µ j = ∇ λ j · v j (cid:54) = 0, j = 1 , , mixed elliptic because thereare necessarily two complex conjugate eigenvalues and one real eigenvalue. HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 11
We compute the approximate eigenvalues of the matrix (36b) in a small amplitudeexpansion and obtain λ = f (cid:48) ( u ) + 116 (cid:32) f (cid:48)(cid:48)(cid:48) ( u ) + ( f (cid:48)(cid:48) ( u )) ( k Ω (cid:48)(cid:48) ( k ) − (cid:48) ( k ))(Ω (cid:48) ( k )) (cid:33) a + O ( a ) , (38a) λ , = f (cid:48) ( u ) + Ω (cid:48) ( k ) ± a (cid:112) − n ( u, k )Ω (cid:48)(cid:48) ( k ) + Λ( u, k ) a + O ( a ) , (38b)where n ( u, k ) is defined in Eq. (24). The O ( a ) correction, Λ( u, k ), to the eigenvaluescorresponding to the split group velocity, λ and λ , is real valued. Consequently,the precise value of this correction does not affect the mathematical structure of thequasilinear system (36a) unless higher order terms are retained. Therefore, we omit theexplicit form of Λ.The asymptotic approximation of the eigenvalues (38) is valid provided that Ω (cid:48) ( k ) (cid:54) = 0.This case will be addressed independently later in this section. We approximate thecorresponding right eigenvectors as(39) v = − Ω (cid:48) ( k ) k f (cid:48)(cid:48) ( u ) + v ( u, k ) v ( u, k )0 a + O ( a ),where(40) v ( u, k ) = v , ∗ ( u, k )16 f (cid:48)(cid:48) ( u )( k Ω (cid:48) ( k )) ,(41) v ( u, k ) = f (cid:48)(cid:48) ( u )(2Ω (cid:48) ( k ) − k Ω (cid:48)(cid:48) ( k )) k Ω (cid:48) ( k ) ,with(42) v , ∗ ( u, k ) = f (cid:48)(cid:48) ( u ) k ( − (cid:48) ( k ) + k Ω (cid:48)(cid:48) ( k ))+ Ω (cid:48) ( k ) ( f (cid:48)(cid:48)(cid:48) ( u ) f (cid:48)(cid:48) ( u ) k − f (cid:48)(cid:48) ( u ) ω ( u, k ) + 16( − kf (cid:48)(cid:48) ( u ) ω ,k ( u, k ) + ω ,u ( u, k )Ω (cid:48) ( k )))+ 16 kω ( u, k )Ω (cid:48) ( k )Ω (cid:48)(cid:48) ( k ) . The remaining eigenvectors are v , = ± (cid:115) − Ω (cid:48)(cid:48) ( k )Ω (cid:48) ( k ) n ( u, k ) f (cid:48)(cid:48) ( u )4Ω (cid:48) ( k )0 a + V ( u, k ) a + O ( a ) , (43)where the O ( a ) correction, V ( u, k ), to the right eigenvectors corresponding to the splitgroup velocity, λ and λ is real valued.We observe that the quasilinear system (36a) is strictly hyperbolic if n ( u, k )Ω (cid:48)(cid:48) ( k ) < n ( u, k )Ω (cid:48)(cid:48) ( k ) > λ and λ given in Eq. (38) are complex conjugates. In this case,the weakly nonlinear Whitham modulation equations (20) are mixed elliptic. The level curves in the u - k plane across which n ( u, k )Ω (cid:48)(cid:48) ( k ) changes sign correspond to certainphysical mechanisms • Ω (cid:48)(cid:48) ( k ) = 0: An extremum of the group velocity • Ω (cid:48) ( k ) = 0, k (cid:54) = 0: short-long wave resonance due to coincident group velocity ω k = f (cid:48) ( u ) + Ω (cid:48) ( k ) and long wavelength phase velocity lim k → ω ( k ) k = f (cid:48) ( u ). • k ) = Ω(2 k ): second harmonic resonance due to coincident phase velocities ofthe first and second harmonics.All of these mechanisms are independent of the mean u . The remaining possibility isthe level curve f (cid:48)(cid:48) ( u ) (2Ω( k ) − Ω(2 k ) + k Ω (cid:48) ( k )) + 12 f (cid:48)(cid:48)(cid:48) ( u )Ω (cid:48) ( k ) (2Ω( k ) − Ω(2 k )) = 0.(45)A calculation shows that if the modulation equations (20) are strictly hyperbolic, allcharacteristic families are genuinely nonlinear if f (cid:48)(cid:48) ( u ) (cid:54) = 0,(46)which is the condition that appears in Corollary 2.At the short-long wave resonance corresponding to Ω (cid:48) ( k ) = 0, B has 1 eigenvaluewith algebraic multiplicity 3 at leading order. The asymptotic approximation of theeigensystem is correspondingly modified and we compute the distinct eigenvalues andcorresponding right eigenvectors(47) λ j = f (cid:48) ( u ) + | kf (cid:48)(cid:48) ( k ) Ω (cid:48)(cid:48) ( k ) | / e jπi/ a / + O ( a / ),(48) v j = + | kf (cid:48)(cid:48) ( k ) Ω (cid:48)(cid:48) ( k ) | / e jπi/ a / + O ( a / ),where j = 0 , ,
2. Two eigenvalues defined in (47) are necessarily complex so the modula-tion system (36a) is mixed elliptic at the short-long wave resonance condition Ω (cid:48) ( k ) = 0.When Ω (cid:48)(cid:48) ( k ) = 0, the modulation system is weakly hyperbolic at O ( a ). A higher orderanalysis is required in order to determine the equation type. If 2Ω( k ) = Ω(2 k ), furtheranalysis is required to determine the system type because the Stokes expansion (35)requires modification so that both the first and second harmonic modes appear at O ( a ).4. Applications to modulational instability
In this section, we discuss the modulational stability/instability of periodic wavetrainsthat can be inferred from the Whitham modulation equations. For a class of HamiltonianPDEs, weak hyperbolicity of the modulation equations is necessary for the modulationalstability of the underlying periodic wave [8]. In the weakly nonlinear regime, additionaldetails of the instability are determined by studying the Nonlinear Schr¨odinger (NLS)equation that describes the slowly evolving nonlinear wave envelope (recall Corollary3). In this section, we demonstrate that the NLS approximation to the generalizedWhitham equation (5) is focusing–which implies the periodic wave is modulationallyunstable–precisely when the weakly nonlinear Whitham modulation equations (20) aremixed elliptic.
HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 13
Nonlinear Schr¨odinger equation approximation.
The NLS equation may bederived from Eq. (5) upon seeking a slowly varying complex wave envelope of the form[54, 1] u = u + (cid:15) (cid:2) Ae iθ + c . c . (cid:3) + (cid:15) (cid:20) f (cid:48)(cid:48) ( u )Ω (cid:48) ( k ) | A | + kf (cid:48)(cid:48) ( u )2Ω( k ) − Ω(2 k ) A e iθ + c . c . (cid:21) + O ( (cid:15) ) , (49)where 0 < (cid:15) = a (cid:28) ξ = (cid:15) ( x − ω (cid:48) ( k ) t ) , τ = (cid:15) t, and find that the complex wave envelope, A ( ξ, τ ), is described to leading order by thecubic NLS equation iA τ + n ( u, k ) | A | A + Ω (cid:48)(cid:48) ( k )2 A ξξ = 0(50)where n is defined exactly as in Eq. (24), which is provided again for reference n ( u, k ) = − k (cid:18) ( f (cid:48)(cid:48) ( u )) Ω (cid:48) ( k ) + k ( f (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) + 12 f (cid:48)(cid:48)(cid:48) ( u ) (cid:19) . (24)The NLS equation (50) is defocusing if n ( u, k )Ω (cid:48)(cid:48) ( k ) < n ( u, k )Ω (cid:48)(cid:48) ( k ) >
0. The competition between nonlinearity and dispersion leads to MI when the NLS equa-tion (50) is focusing. As expected, the NLS approximation to Eq. (5) is focusing precisely when the weakly nonlinear Whitham modulation equations (20) are mixed elliptic, asdefined by the criterion given in Eq. (44). When n ( u, k )Ω (cid:48)(cid:48) ( k ) >
0, the focusing NLSequation predicts the growth rate of small amplitude perturbations to a steady periodictraveling wave. The growth rate is identified by seeking a solution of Eq. (50) of theform A = e in ( u,k ) τ (cid:0) αe iκξ + γτ (cid:1) , (51)where α is the small, complex perturbation amplitude. We linearize about the stationarysolution to find the relation between κ and γ . Instability occurs when Re γ >
0, where γ = κ (cid:115) n ( u, k )Ω (cid:48)(cid:48) ( k ) − (Ω (cid:48)(cid:48) ( k )) κ , (52)which is real for perturbation wavenumbers 0 < κ < n ( u,k )Ω (cid:48)(cid:48) ( k ) . The maximal growth rateof the instability occurs when κ = κ max κ = n ( u, k )Ω (cid:48)(cid:48) ( k )(53)where the corresponding maximal growth rate is given by γ max = | n ( u, k ) | . (54)Note that the NLS equation (50) is valid when n ( u, k )Ω (cid:48)(cid:48) ( k ) (cid:54) = 0. At an inflection pointof the dispersion, a higher order NLS approximation is required where an alternativecriteria for MI can be obtained [3]. The remainder of this section is dedicated to studying the modulational stability of pe-riodic wavetrains in several concrete physical systems. We will use the index n ( u, k )Ω (cid:48)(cid:48) ( k )to determine curves in the u - k plane where wavetrains transition from stable to unstable.The only requirements for this analysis are the nonlinear flux and linear dispersion forthe generalized Whitham equation (5). Table 1 summarizes these nonlinear fluxes anddispersion relations for three applications. Relevant parameters will be defined in thesubsequent sections when we present the stability results.Application f ( u ) Ω( k )Gravity-capillarywater waves u − u (cid:112) ( k + Bk ) tanh( k ) − k Hydroelastic waterwaves u − u (cid:112) ( k + Dk ) tanh k − k Internal waves α u + α u (cid:113) k (1 − ˜ ρ )˜ ρ coth( k )+coth( k ˜ h − ) − ck Table 1.
Nonlinear flux and linear dispersion relation in the generalizedWhitham equation (5) for various problems.4.2.
Gravity-capillary water waves.
As mentioned at the beginning of this section,the study of modulational instability of periodic water waves propagating on a finitedepth dates back to the work of Benjamin [4] and Whitham [52], who, using independentmethods, discovered the same transition from modulational stability to instability as theproduct of the undisturbed fluid depth, h , to the periodic wave’s wavenumber k passesthrough a critical threshold. For surface gravity waves, the threshold is kh = η cr , (55)where η cr ≈ . kh < η cr , weakly nonlinear water waves are stable. Otherwise,they are unstable. In the subsequent discussion, equations are cast in nondimensionalform (see ref. [26]) such that h = 1.For pure gravity waves and weak nonlinearity, the nonlinear flux and dispersion are(cf. Eq. (2)) f ( u ) = 34 u , Ω( k ) = √ k tanh k − k, (56)which yields the original Whitham equation (up to a Gallilean transformation) with thestandard nondimensionalization so that the undisturbed fluid depth is unity. Small, butfinite amplitude periodic solutions to the Whitham equations were proven to be unstableas the nondimensional wavenumber passes through k ≈ . < η cr [24, 41]. The index n ( u, k )Ω (cid:48)(cid:48) ( k ) exactly reproduces this result.Subsequent work incorporated surface tension into the Whitham equation, which ap-pears solely in the linear dispersion relation. The simplest model of gravity capillary HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 15 waves with full dispersion is the Whitham equation with quadratic nonlinear flux (56)and the linear dispersion relation(57) Ω( k ) = (cid:112) ( k + Bk ) tanh k − k, where B , the Bond number, is a measure of the strength of the surface tension force inrelation to the gravitational force. When B = 0, this model reduces to the Whithamequation for gravity water waves. The presence of surface tension results in a varietyof novel phenomena, particularly near the critical value B = 1 /
3, where short wavedispersion prevails over long wave dispersion. Studies of the generalized Whitham equa-tion (5) with dispersion (57) and quadratic nonlinear flux (56) established regions in the( k, B ) parameter space for which weakly nonlinear periodic traveling wave solutions aremodulationally unstable [23, 10].We now incorporate a higher order nonlinear flux term into the generalized Whithamequation (5). A convenient method to obtain the flux was proposed by Whitham (see[52] p. 478). We retain nonlinear terms up to third order in u and consider the gener-alized Whitham equation (5) defined by the nonlinear flux and linear dispersion givenin Table 1. We compute the modulational instability index n (0 , k )Ω (cid:48)(cid:48) ( k ) with u = 0and compare the results to known results for the full water wave problem [28] in Fig.2. Regions of ( k, B ) parameter space where periodic waves are modulationally stableappear in gray whereas regions that correspond to modulationally unstable waves areshown in white. The generalized Lighthill-Whitham criterion for modulational insta-bility improves upon the quadratic flux results [23] and is in good agreement with thestability analysis of solutions of the Euler equations with surface tension. The five reddashed curves correspond to a calculation of MI for the full Euler equations. The lowestsolid curve intersects the k -axis where the growth rate becomes real (unstable), whichapproximates the cutoff for the Benjamin-Feir instability (55). As the nondimensionalwavenumber passes through approximately k ≈ . k ≈ .
146 [23, 41]). The MI index only includesderivatives of f ( u ) up to third order evaluated at u = 0, so agreement with the stabilityanalysis of the Euler equations will not be improved by including additional polynomialnonlinear terms. The three middle curves demarcating the stable/unstable transitionmatch exactly with those from the Euler equation since they depend only on the lineardispersion relation [23]. This dependence is specified in Fig. 2.4.3. Hydroelastic water waves.
Hydroelastic, or flexural-gravity water waves referto waves at the free surface of a body of water bounded by a deformable, elastic thinsheet. Physically, this may refer to the configuration where an inviscid, irrotational fluidis underneath a relatively thin ice sheet that is assumed to cause no friction with thewater wave surface and modeled by an additional pressure term in the dynamic freesurface boundary condition of the Euler equations [39]. Of particular interest in systemsof this type is the influence of a moving, localized forcing. Experimental observation ofmoving vehicles over a frozen body of water demonstrate that typically a wide rangeof wavelengths are observed while the amplitude of the oscillations remains small [44]. ⌦ ( k ) = ⌦ ( k ) = 0 k ) = ⌦(2 k ) Figure 2.
Evaluation of the MI index n (0 , k )Ω (cid:48)(cid:48) ( k ) for gravity-capillarywater waves modeled by a generalized Whitham equation with f ( u ) andΩ( k ) given in Table 1. Gray regions correspond to positive MI indexwaves (stable) and white regions coincide with negative MI index (un-stable). Black, solid curves demarcate the border between the regions ofstability/instability. The dashed red curves indicate a change in the sta-bility of a finite amplitude periodic traveling wave solution of the fullynonlinear Euler equations [28].However, nonlinear effects, albeit weak, likely play a role in the wave evolution andvarious works demonstrate that nonlinearity has a marked effect on hydroelastic wavedynamics both for localized disturbances [40, 47] and periodic waves [45, 53]. Recently,the Whitham equation with quadratic nonlinearity was investigated in this context tocapture weak nonlinearity and full linear wave dispersion [14] capturing qualitative fea-tures of solutions computed in the aforementioned studies.The primary effect of the rigid ice sheet is to modify the linear dispersion, but notthe nonlinear flux of free surface water waves [53]. The generalized Whitham equationwe propose here includes flux terms up to third order and linear dispersion defined inTable 1. The parameter D > n (0 , k )Ω (cid:48)(cid:48) ( k )indicates modulationally stable or unstable periodic solutions with zero mean. In Fig.3, we have chosen axes to compare with Fig. 2.4.4. Internal waves.
We now consider the stability of internal waves for two immisciblefluid layers with differing densities. We identify the physical configuration where anincompressible fluid of density ρ lies above a finite fluid layer of density ρ with ρ > ρ to maintain a stable stratification. The undisturbed interface between the two fluidsat the origin of the vertical coordinate z = 0 is bounded above by an impenetrableboundary at z = + h and below by a similar boundary at z = − h . A sketch of theconfiguration considered here is shown in Fig. 4 HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 17 ⌦ ( k ) = ⌦ ( k ) = 0 k ) = ⌦(2 k ) Figure 3.
Evaluation of the MI index n (0 , k )Ω (cid:48)(cid:48) ( k ) for flexural waterwaves modeled by a generalized Whitham equation with f ( u ) and Ω( k )given in Table 1. Gray regions correspond to positive MI index waves(stable) and white regions coincide with negative MI index (unstable).Black solid curves demarcate the border between the regions of stabil-ity/instability. xz z = 0 z = h z = h u ( x, t ) ⇢ ⇢ Figure 4.
Configuration for internal water waves at the free interface oftwo stratified fluids.A long wave asymptotic expansion of the two fluid system when h → ∞ yields theBenjamin equation [5], which consists of the KdV third order dispersion term and aHilbert transform term identical to that in the Benjamin-Ono equation. Other limitsrecover the KdV and intermediate-long wave equations [11]. Internal waves observedin the ocean are known to exhibit strong nonlinearity [21]. Larger amplitude internalwaves can be modeled by the Gardner equation with both quadratic and cubic nonlinearflux terms [15]. Here, we retain up to cubic nonlinear terms and full linear dispersion,mirroring the previous two sections. In this physical system, we nondimensionalizecoordinates according to x → h ˜ x, t → (cid:115) h g ˜ t, u → h ˜ u, (58) where g is the gravitational acceleration. The nonlinear flux and linear dispersion of thegeneralized Whitham equation (5) in nondimensional coordinates (upon dropping tildesfrom variables) are f ( u ) = α u + α u , (59a) Ω( k ) = (cid:115) k (1 − ˜ ρ )˜ ρ coth k + coth k ˜ h − ck, (59b)where α = 32 c (˜ h − ˜ ρ )( ˜ ρ + ˜ h ) , (59c) α = c (cid:32) ˜ ρ − ˜ h ˜ ρ + ˜ h (cid:33) − ˜ ρ + ˜ h ˜ ρ + ˜ h , (59d) c = (cid:115) (1 − ˜ ρ )˜ ρ + ˜ h , (59e) ˜ ρ = ρ ρ , (59f) ˜ h = h h . (59g)Note that the dispersion and nonlinear flux now depend on the two parameters ˜ ρ and ˜ h .We now compute examples of regions corresponding to stable/unstable waves for twofixed density ratios. We focus on the modulational stability of a range of wavelengthsby plotting on a log-log scale for the two density ratios ˜ ρ = 0 . ρ = 0 .
99 shownin Fig. 5. Qualitatively, the curves separating the parameter regions corresponding tostable/unstable periodic waves are similar, though there are minor quantitative differ-ences. It is not surprising that long waves are modulationally stable since this is theregion where the KdV or Benjamin-Ono apply. Note that the upper and lower boundsof the region of stable periodic waves approach ˜ h = √ ˜ ρ as k → ∞ . This coincides withparameter values for which the coefficient of the quadratic nonlinear (cf. Eq. (59c))term vanishes. HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 19
Figure 5.
Evaluation of the index n (0 , k )Ω (cid:48)(cid:48) ( k ) for waves at a two fluidinterface modeled by a generalized Whitham equation with f ( u ) and Ω( k )given in (59) for two fixed density ratios ˜ ρ . Gray regions correspond topositive MI index (stable) and white regions coincide with negative MI in-dex (unstable). Black solid curves mark the border of the stable/unstableregions with density ratio ˜ ρ = 0 .
99. The dashed red curves are the divisionof the stable/unstable regions for the density ratio ˜ ρ = 0 . Discussion and conclusions
In this manuscript, we derived the Whitham modulation equations for nonlinear,fully dispersive scalar model equations we called the generalized Whitham equation (5).The presence of the nonlocal dispersive term in this model motivated a Fourier basedmultiple scale approach to derive the modulation equations. The modulation equationsare general but concrete, involving averages over the periodic traveling wave manifold.We then considered the Whitham modulation equations in a weakly nonlinear regime,where periodic traveling wave solutions to the generalized Whitham equation were ob-tained. In so doing, we determined the modulation equations in explicit, quasilinearform. From there, we derived conditions under which the system was strictly hyperbolicor mixed elliptic and genuinely nonlinear.Finally, we considered the application of this theory to modulational instability. Byderiving a criterion for hyperbolicity of the modulation equations in the weakly nonlinearregime, we obtained an explicit generalized Lighthill-Whitham criterion for modulationalinstability that depends entirely on the nonlinear flux f ( u ) and linear dispersion relationΩ( k ). We applied this MI index to various geophysical fluid applications: gravity-capillary water waves, hydroelastic water waves, and internal waves.Our results are applicable to many physical models by simply specifying the nonlinearflux f ( u ) and linear dispersion relation Ω( k ), subject to modest assumptions. Moreover,we derived the NLS equation in the weakly nonlinear regime from the generalized modelEq. (5). The nonlinear coefficient n ( u, k ) is found by selecting the nonlinear flux f ( u ) and linear dispersion relation Ω( k ) in Eq. (24), as opposed to full asymptotic re-derivationfor different nonlinear fluxes. As such, the results given in this work are of particularinterest to those with applications where full dispersion scalar models are used.There are several potential directions for future research. While we analyzed themodulation equations in the weakly nonlinear regime, an analysis of the modulationequations in the fully nonlinear regime is of interest as well. More generally, the easeof applying the results of Corollary 2 to general full dispersion scalar models meansthat this work could be extended to applications beyond those we discussed in Section4. Explicit expressions for families of periodic solutions are not always available, soproperties of the modulation equations can be determined numerically. These numericalcomputations have been carried out for the conduit equation [36] and the fifth orderKdV equation [43].The Whitham modulation equations (15) can be used to study dynamical solutions ofcertain initial value problems. A common application of Whitham modulation theory isto study the structure of dispersive shock waves (DSWs), which typically consist of anexpanding, modulated periodic wave with a harmonic wavetrain at one edge and an ap-proximate solitary wave at the opposite edge. The Whitham modulation equations canbe used to determine both macroscopic properties, e.g. the velocities of the disparateedges, and the DSW’s interior structure [20]. Novel dynamics in scalar models contain-ing cubic nonlinear flux [19] or higher order/nonlocal dispersion [42, 20, 22] have beenobserved. These include so-called Whitham shocks–discontinuous weak solutions of themodulation equations that correspond to zero dispersion limits of periodic heteroclinicto periodic traveling wave solutions [43]. A promising research direction is the consid-eration of Whitham shocks for the generalized Whitham equation (5). The derivationof the general NLS equation with nonlinear coefficient n ( u, k ) given in Eq. (24) can beused for a universal description of the DSW structure near the harmonic edge [12]. Thegeneral NLS equation presented in this manuscript allows for this approach to be usedfor any model equation that can be cast in the form of Eq. (5).Finally, the Fourier based approach used to derive the Whitham modulation equationscan be applied to study more general full dispersion equations. These may includemodels incorporating nonlinear dispersive terms (for instance of Camassa-Holm type[29]), Boussinesq systems [25, 2], or full dispersion NLS models [38]. Acknowledgments
This work was partially supported by NSF grant DMS-1816934.
References
1. M. J. Ablowitz,
Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons , Cambridge Uni-versity Press, 2011.2. P. Aceves-Snchez, A.A. Minzoni, and P. Panayotaros,
Numerical study of a nonlocal model forwater-waves with variable depth , Wave Motion (2013), 80–93.3. S. Amiranashvili and E. Tobisch, Extended criterion for the modulation instability , New Journal ofPhysics (2019), 033029.4. T. B. Benjamin, Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discus-sion] , Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences (1967), 59–76.
HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 21
5. ,
A new kind of solitary wave , Journal of Fluid Mechanics (1992), 401.6. T. B. Benjamin and J. E. Feir,
The Disintegration of Wave Trains on Deep Water Part 1. Theory ,Journal of Fluid Mechanics Digital Archive (1967), 417–430.7. S. Benzoni-Gavage, C. Mietka, and L. M. Rodrigues, Modulated equations of Hamiltonian PDEsand dispersive shocks , arXiv:1911.10067 [math] (2019).8. S. Benzoni-Gavage, P. Noble, and L. M. Rodrigues,
Slow Modulations of Periodic Waves in Hamil-tonian PDEs, with Application to Capillary Fluids , Journal of Nonlinear Science (2014), 711–768.9. J. D. Carter, Bidirectional Whitham equations as models of waves on shallow water , Wave Motion (2018), 51–61.10. J. D. Carter and M. Rozman, Stability of Periodic, Traveling-Wave Solutions to theCapillaryWhitham Equation , Fluids (2019), 58.11. W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system , Journal of FluidMechanics (1999), 1–36.12. T. Congy, G.A. El, M.A. Hoefer, and M. Shearer,
Nonlinear Schrdinger equations and the universaldescription of dispersive shock wave structure , Studies in Applied Mathematics (2019), 241–268.13. C. M. Dafermos,
Hyperbolic Conservation Laws in Continuum Physics , 4th ed., Springer, Berlin,2016.14. E. Dinvay, H. Kalisch, D. Moldabayev, and E. I. P˘ar˘au,
The Whitham equation for hydroelasticwaves , Applied Ocean Research (2019), 202–210.15. V. D. Djordjevic and L. G. Redekopp, The Fission and Disintegration of Internal Solitary WavesMoving over Two-Dimensional Topography , Journal of Physical Oceanography (1978), 1016–1024.16. M. Ehrnstr¨om, M. D. Groves, and E. Wahl´en, On the existence and stability of solitary-wave solu-tions to a class of evolution equations of Whitham type , Nonlinearity (2012), 2903–2936.17. M. Ehrnstr¨om and H. Kalisch, Traveling waves for the Whitham equation , Differential and IntegralEquations (2009), 1193–1210.18. , Global Bifurcation for the Whitham Equation , Mathematical Modelling of Natural Phe-nomena (2013), 13–30.19. G. El, M. Hoefer, and M. Shearer, Dispersive and Diffusive-Dispersive Shock Waves for NonconvexConservation Laws , SIAM Review (2017), 3–61.20. G. A. El, L. T. K. Nguyen, and N. F. Smyth, Dispersive shock waves in systems with nonlocaldispersion of Benjamin-Ono type , Nonlinearity (2018), 1392–1416.21. K. R. Helfrich and W. K. Melville, Long nonlinear internal waves , Annual Review of Fluid Mechanics (2006), 395–425.22. M. A. Hoefer, N. F. Smyth, and P. Sprenger, Modulation theory solution for nonlinearly resonant,fifth-order Korteweg-de Vries, nonclassical, traveling dispersive shock waves , Studies in AppliedMathematics (2019), 219–240.23. V. M. Hur and M. A. Johnson,
Modulational Instability in the Whitham Equation for Water Waves ,Studies in Applied Mathematics (2015), 120–143.24. ,
Modulational instability in the Whitham equation with surface tension and vorticity , Non-linear Analysis: Theory, Methods & Applications (2015), 104–118.25. V. M. Hur and A. K. Pandey,
Modulational instability in a full-dispersion shallow water model ,Studies in Applied Mathematics (2019), 3–47.26. R. S. Johnson,
A Modern Introduction to the Mathematical Theory of Water Waves , CambridgeTexts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.27. R. I. Joseph,
Solitary waves in a finite depth fluid , Journal of Physics A: Mathematical and General (1977), L225–L227.28. T. Kawahara, Nonlinear Self-Modulation of Capillary-Gravity Waves on Liquid Layer , Journal ofthe Physical Society of Japan (1975), 265–270.29. D. Lannes, The water waves problem , American Mathematical Society, 2013.30. P. D. Lax,
Hyperbolic systems of conservation laws and the mathematical theory of shock waves ,SIAM, 1973.
31. P. D. Lax and C. D. Levermore,
The small dispersion limit of the Korteweg-de Vries equation: 1 ,Comm. Pure Appl. Math. (1983), 253–290.32. , The small dispersion limit of the Korteweg-de Vries equation: 2 , Comm. Pure Appl. Math. (1983), 571–593.33. , The small dispersion limit of the Korteweg-de Vries equation: 3 , Comm. Pure Appl. Math. (1983), 809–830.34. M. J. Lighthill, Contributions to the Theory of Waves in Non-linear Dispersive Systems , IMAJournal of Applied Mathematics (1965), 269–306.35. J. C. Luke, A perturbation method for nonlinear dispersive wave problems , Proceedings of the RoyalSociety of London. Series A. Mathematical and Physical Sciences (1966), 403–412.36. M. D. Maiden and M. A. Hoefer,
Modulations of viscous fluid conduit periodic waves , Proc. R. Soc.A (2016), 20160533.37. D. Moldabayev, H. Kalisch, and D. Dutykh,
The Whitham Equation as a model for surface waterwaves , Physica D: Nonlinear Phenomena (2015), 99–107.38. P. I. Naumkin and I. A. Shishmarev,
Nonlinear Nonlocal Equations in the Theory of Waves , vol.133, Amer Mathematical Society, 1994.39. P. I. Plotnikov and J. F. Toland,
Modelling nonlinear hydroelastic waves , Philosophical Transactionsof the Royal Society A: Mathematical, Physical and Engineering Sciences (2011), 2942–2956.40. E. P˘ar˘au and F. Dias,
Nonlinear effects in the response of a floating ice plate to a moving load ,Journal of Fluid Mechanics (2002), 281–305.41. N. Sanford, K. Kodama, J. D. Carter, and H. Kalisch,
Stability of traveling wave solutions to theWhitham equation , Physics Letters A (2014), 2100–2107.42. P. Sprenger and M. Hoefer,
Shock Waves in Dispersive Hydrodynamics with Nonconvex Dispersion ,SIAM Journal on Applied Mathematics (2017), 26–50.43. P. Sprenger and M. A. Hoefer, Discontinuous shock solutions of the Whitham modulation equationsas zero dispersion limits of traveling waves , Nonlinearity (2020), 3268–3302.44. T. Takizawa, Field Studies on Response of a Floating Sea Ice Sheet to a Steadily Moving Load ,Journal of Geophysical Research (1988), 47.45. O. Trichtchenko, P. Milewski, E. P˘ar˘au, and J.-M. Vanden-Broeck, Stability of periodic travelingflexural-gravity waves in two dimensions , Studies in Applied Mathematics (2019), 65–90.46. S. Trillo, M. Klein, G.F. Clauss, and M. Onorato,
Observation of dispersive shock waves developingfrom initial depressions in shallow water , Physica D: Nonlinear Phenomena (2016), 276–284.47. J.-M. Vanden-Broeck and E. I. P˘ar˘au,
Two-dimensional generalized solitary waves and periodicwaves under an ice sheet , Philosophical Transactions of the Royal Society A: Mathematical, Physicaland Engineering Sciences (2011), 2957–2972.48. S. Venakides,
The zero-dispersion limit of the Korteweg-de Vries equation with non-trivial reflectioncoefficient , Comm. Pure Appl. Math. (1985), 125–155.49. G. B. Whitham, A general approach to linear and non-linear dispersive waves using a Lagrangian ,Journal of Fluid Mechanics (1965), 273–283.50. , Non-linear dispersive waves , Proc. Roy. Soc. Ser. A (1965), 238–261.51. ,
Variational methods and applications to water waves , Proceedings of the Royal Society ofLondon. Series A. Mathematical and Physical Sciences (1967), 6–25.52. ,
Linear and nonlinear waves , Wiley, New York, 1974.53. X. Xia and H. T. Shen,
Nonlinear interaction of ice cover with shallow water waves in channels ,Journal of Fluid Mechanics (2002), 259–268.54. V. E. Zakharov and E. A. Kuznetsov,
Multi-scale expansions in the theory of systems integrable bythe inverse scattering transform , Physica D: Nonlinear Phenomena (1986), 455–463.55. V.E. Zakharov and L.A. Ostrovsky, Modulation instability: The beginning , Physica D: NonlinearPhenomena (2009), 540–548.
HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 23
Appendix A. Multiple scale expansion of the nonlocal convolutionoperator
The multiple scale expansion of the convolution operator in Sec. 3 was requiredto derive the Whitham modulation equations (15). In this appendix we derive thisexpansion.The modulation equations are derived upon assuming the existence of a periodic wavewith the slowly varying ansatz u ( x, t ) = ϕ ( θ, X, T ) ≡ ϕ ( θ ; p ( X, T )) X = (cid:15)x, T = (cid:15)t, < (cid:15) (cid:28) , (60)in which the vector of parameters p varies on the slow scales X and T while ϕ remains2 π -periodic in θ : ϕ ( θ + 2 π, X, T ) = ϕ ( θ, X, T ), for all θ, X ∈ R , T >
0. This multiscaleansatz (60) results in the usual differential operator expansions(61) ∂ x = k∂ θ + (cid:15)∂ X , ∂ t = − ω∂ θ + (cid:15)∂ T . The ansatz (60) admits the two-scale Fourier series-transform representation ϕ ( θ, X, T ) = 12 π ∞ (cid:88) n = −∞ (cid:90) R (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q, (cid:98) ϕ n ( Q, T ) = 12 π (cid:90) π − π (cid:90) R ϕ ( θ, X, T ) e − i ( nθ + QX ) d X d θ, (62)where integers n are harmonics of the rapidly varying, locally periodic wave and Q isthe modulation wavenumber. The convolution operator can then be interpreted in theFourier domain (cid:92) K ∗ ϕ x = (cid:90) R (cid:20) k ∂∂θ + (cid:15) ∂∂X (cid:21) (cid:32) Ω( q ) q (cid:34) π ∞ (cid:88) n = −∞ (cid:90) R (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q (cid:35)(cid:33) e − iqx d x (63) = (cid:90) R (cid:88) n i [ nk + (cid:15)Q ] Ω( q ) q (cid:98) ϕ n ( Q, T ) δ ( (cid:15)Q + nk − q ) e iQx d Q, (64)where δ is the Dirac delta. Interchanging the order of integration and summation, wefind that the multiscale expansion of the spatial wavenumber in the Fourier domain is q → nk + (cid:15)Q. (65)For the modulated periodic solutions considered here, the convolution operator is amultiplier on the modulated Fourier modes K ∗ ϕ x = 12 π ∞ (cid:88) n = −∞ (cid:90) R Ω( nk + (cid:15)Q ) nk + (cid:15)Q (cid:98) ϕ n ( Q, T ) i ( nk + (cid:15)Q ) e i ( nθ + QX ) d Q = 12 π ∞ (cid:88) n = −∞ (cid:90) R i Ω( nk + (cid:15)Q ) (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q ∼ π ∞ (cid:88) n = −∞ (cid:90) R i (Ω( nk ) + (cid:15)Q Ω (cid:48) ( nk ) + . . . ) (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q (66) Rearranging terms in Eq. (66), we find
K ∗ ϕ x ∼ π ∞ (cid:88) n = −∞ (cid:90) R i Ω( nk ) (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q + (cid:15)∂ X (cid:32) π ∞ (cid:88) n = −∞ (cid:90) R Ω (cid:48) ( nk ) (cid:98) ϕ n ( Q, T ) e i ( nθ + QX ) d Q (cid:33) . (67)The multiscale expansion of the convolution is therefore K u x ∼ k K ∗ u θ + (cid:15)∂ X ( K (cid:48) ∗ u ) , (68)where the convolution K (cid:48) ∗ u is defined by the Fourier multiplier (cid:92) K (cid:48) ∗ u = Ω (cid:48) ( k ) (cid:98) u. (69)Inserting the series expansion for the modulated periodic traveling wave, Eq. (8) con-cludes the proof of Lemma 1. Appendix B. Stokes Expansion
In this appendix, we compute an approximate periodic, traveling wave solution to Eq.(28) via an asymptotic expansion in the wave amplitude, a . The expansion is of theform(70) ϕ = u + a ϕ + a ϕ + . . . ,(71) ω = ω + aω + a ω + . . . ,where u is the mean and ω is the linear dispersion relation ω ( u, k ) = f (cid:48) ( u ) k + Ω( k ) . (72)We insert the asymptotic expansions (70) and (71) into (28) and collect in powers ofthe amplitude. The leading order problem is trivially satisfied, and the first nontrivialproblem appears at O ( (cid:15) ) where we have(73) − ω ϕ (cid:48) + kf (cid:48) ( u ) ϕ (cid:48) + K ∗ ( kϕ (cid:48) ) = 0 , where primes denote derivatives with respect to θ . The integro-differential equation canbe solved with Fourier transforms. The 2 π periodic solution at leading order is given by ϕ = cos θ. We continue to the next order O ( a ) where we have the intego-differential equation for ϕ (74) − ω ϕ (cid:48) + kf (cid:48) ( u ) ϕ (cid:48) + K ∗ ( kϕ (cid:48) ) = ω ϕ (cid:48) − kf (cid:48)(cid:48) ( u ) ϕ ϕ (cid:48) . The linear equation is solvable provided the forcing terms are orthogonal to the kernelof the linear operator that defines the integro-differential equation. Secular terms areremoved so long as ω = 0. We solve Eq. (74) with Fourier transforms and find(75) ϕ = kf (cid:48)(cid:48) ( u )4Ω( k ) − k ) sin(2 θ ) HITHAM MODULATION THEORY FOR GENERALIZED WHITHAM EQUATIONS 25
The O ( a ) correction to the frequency is found by ensuring no secular terms appear at O ( (cid:15) ) − ω ϕ (cid:48) + kf (cid:48) ( u ) ϕ (cid:48) + K ∗ ( kϕ (cid:48) )= ω ϕ (cid:48) − kf (cid:48)(cid:48) ( u ) ϕ ϕ (cid:48) − kf (cid:48)(cid:48) ( u ) ϕ ϕ (cid:48) − kf (cid:48)(cid:48)(cid:48) ( u )2 ϕ ϕ (cid:48) . (76)Secular terms–coefficients of cos( θ ) terms–are removed upon setting(77) ω ( k ) = k (cid:32) k ( f (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) + f (cid:48)(cid:48)(cid:48) ( u )2 (cid:33) .Therefore, the approximation for the weakly nonlinear periodic solution is given by(78) u = u + a θ + kf (cid:48)(cid:48) ( u )16Ω( k ) − k ) cos 2 θ + O ( a )with frequency ω ( k, u, a ) = kf (cid:48) ( u ) + Ω( k ) + a k (cid:32) k ( f (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) + f (cid:48)(cid:48)(cid:48) ( u )2 (cid:33) + O ( a ) , (79)where a (cid:28) f ( u ) and Ω( k ). Note that theStokes solution is invalid if 2Ω( k ) = Ω(2 k ), which corresponds to a resonance due tocoincident phase velocities at the first and second harmonic. Appendix C. Derivation of the NLS equation
The Nonlinear Schr¨odinger (NLS) equation describes the weakly nonlinear complexenvelope of a carrier wave. The asymptotic derivation of the NLS equation assumed asolution in a small parameter 0 < (cid:15) = a (cid:28) u = u ( ζ, X, T ) + (cid:15)u ( ζ, X, T ) + (cid:15) u ( ζ, X, T ) + . . . , where X and T are the slow space and time scales introduced previously and ζ = kx − ( f (cid:48) ( u ) k + Ω( k )) t . At O ( (cid:15) ), we obtain the linear, homogeneous equation M u := {− Ω( k ) ∂ ζ + K ∂ ζ } u = 0(80)which has the solution u = A ( X, T ) e iζ + c . c . + M ( X, T ), where c . c . denotes the complexconjugate. At O ( (cid:15) ) we have the nonhomogeneous equation M ( u ) = − u ,T − f (cid:48) ( u ) u ,X − Ω (cid:48) ( − ik∂ ζ ) u ,X − kf (cid:48)(cid:48) ( u ) u u ,ζ , (81)which contains no secular terms provided that A T + ω (cid:48) A X + ikf (cid:48)(cid:48) ( u ) M A = (cid:15)G + (cid:15) G + . . . (82) M T = (cid:15)F + (cid:15) F + . . . , (83)where we have introduced the higher order corrections G i and F i , i = 1 , , . . . . Wesolve Eq. (81) and find u = kf (cid:48)(cid:48) ( u ) A k ) − Ω(2 k ) e iζ + c . c .. (84) We continue the asymptotic procedure to the next order in (cid:15) . At this order, solvabilitywith respect to the first harmonic ( e iζ ) and constant forcing terms result respectively in A T + ω (cid:48) A X + kf (cid:48)(cid:48) ( u ) M A = (cid:15) (cid:18) − i ( kf (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) − i k f (cid:48)(cid:48)(cid:48) ( u ) (cid:19) | A | A + i(cid:15) Ω (cid:48)(cid:48) ( k )2 A XX , (85) M T + f (cid:48) ( u ) M X = − (cid:15)f (cid:48)(cid:48) ( u ) (cid:0) | A | (cid:1) X , (86)which shows that M is O ( (cid:15) ) in the asymptotic expansion. Upon introducing the variables ξ = X − ω (cid:48) ( k ) T, τ = (cid:15)T, we find that, to O ( (cid:15) ), Eq. (86) can be solved to find that M = (cid:15) f (cid:48)(cid:48) ( u )Ω (cid:48) ( k ) | A | . (87)Replacing M in Eq. (85) with the expression (87) we find that the nonlinear envelope, A , solves the NLS equation (cf. Eq. (50)) iA τ + n ( u, k ) | A | A + Ω (cid:48)(cid:48) ( k )2 A ξξ = 0(88)where n ( u, k ) = − k (cid:18) ( f (cid:48)(cid:48) ( u )) Ω (cid:48) ( k ) + k ( f (cid:48)(cid:48) ( u )) k ) − Ω(2 k ) + 12 f (cid:48)(cid:48)(cid:48) ( u ) (cid:19) . E-mail address : [email protected] School of Natural Sciences, University of California Merced, Merced, USA Department of Applied Mathematics, University of Colorado Boulder, Boulder,USA3