Modulated equations of Hamiltonian PDEs and dispersive shocks

Benzoni-Gavage, Sylvie; Mietka, Colin; Rodrigues, Luis Miguel

doi:10.1088/1361-6544/abcb0a

Cited by 15 publications

(27 citation statements)

References 46 publications

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“…Equivalently, they describe the slow evolution of a nonlinear, periodic wavetrain's parameters. Various methods exist to derive the Whitham modulation equations including averaged conservation laws, 19 averaged Lagrangian, 20 averaged Hamiltonian, 21 or a multiple scale procedure 22 . For the KdV equation (), the Whitham equations were proven to describe the zero dispersion limit 23–26 for

L^{2} false(double-struckR false)

data with the inverse scattering transform.…”

Section: Introductionmentioning

confidence: 99%

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

et al. 2021

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The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity ffalse(ufalse)=12u2 of the Korteweg–de Vries equation and the full linear dispersion relation Ωfalse(kfalse)=ktanhk of unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux function f(u) and a general linear dispersion relation Ω(k). Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling 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 hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability.

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L^{2} false(double-struckR false)

data with the inverse scattering transform.…”

Section: Introductionmentioning

confidence: 99%

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

et al. 2021

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“…We see this as a welcome complement to the development of numerical tools for 'arbitrary' periodic waves in the bulk [2,16], in particular because these tools can hardly attain asymptotic regimes or cover the whole range of parameters. Even more importantly, the asymptotic expansions derived here in the small amplitude and solitary wave limits to elucidate stability conditions are per se involved in many related issues [5,4,3], including the understanding of dispersive shocks to be investigated elsewhere. An initially unexpected outcome of this work is that it fills the gap in between the stability conditions mentioned above, in those asymptotic regimes.…”

mentioning

confidence: 99%

“…In a companion paper [3], we consider the modulated equations associated with our class of Hamiltonian PDEs, and use the asymptotic expansions derived here to gain insight on modulated equations in the small amplitude and in the soliton limits, which are crucial to the understanding of dispersive shocks. In further work, including the forthcoming [6] focused on scalar equations, we plan to also address the joint limit and the existence of small amplitude dispersive shocks for Hamiltonian PDEs.…”

mentioning

confidence: 99%

Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude

Benzoni-Gavage¹,

Mietka²,

Rodrigues³

2020

Indiana Univ. Math. J.

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Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature -or Morse index -of the Hessian matrix of action integrals associated with those waves. In a previous paper (published in Nonlinearity in 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg-de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by M (c), the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave. This gives an alternate proof of Gardner's result [J. Reine Angew. Math. 1997] according to which the instability of the limiting solitary wave, when M (c) is negative, implies the instability of nearby periodic waves. Interestingly enough, it is found here that in the inverse situation, when M (c) is positive, nearby periodic waves are orbitally stable with respect to co-periodic perturbations.

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“…Still, even if the equations can now be explicitly written in terms of a, u, m, it is difficult to extract from ( 23)-( 25) "reasonably simple" closed form solutions to compare with numerical solutions of the exact BBM equation. The idea is to find the solitary limit of ( 23), ( 24), ( 25) [17,6]. This limit is singular, and we cannot obtain this limit directly from the three above written conservation laws.…”

Section: Whitham Modulation Equations For the Bbm Systemmentioning

confidence: 99%

“…The Riemann problem for the BBM equation is the Cauchy problem u(0, x) = u − , x < 0, u + , x > 0. (6) with constant values of u ± . Such a problem is often called Gurevich-Pitaevskii problem, who were the first to give its asymptotic solution for the Kortewegde Vries (KdV) equation [20].…”

Section: Introductionmentioning

confidence: 99%

Singular solutions of the BBM equation: analytical and numerical study

Gavrilyuk

Shyue

2021

Nonlinearity

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We show that the Benjamin–Bona–Mahony (BBM) equation admits stable travelling wave solutions representing a sharp transition from a constant state to a periodic wave train. The constant state is determined by the parameters of the periodic wave train: the wave length, amplitude and phase velocity, and satisfies both the generalized Rankine–Hugoniot conditions for the exact BBM equation and for its wave averaged counterpart. Such stable shock-like travelling structures exist if the phase velocity of the periodic wave train is not less than the solution wave averaged. To validate the accuracy of the numerical method, we derive the (singular) solitary limit of the Whitham system for the BBM equation and compare the corresponding numerical and analytical solutions. We find good agreement between analytical results and numerical solutions.

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Modulated equations of Hamiltonian PDEs and dispersive shocks

Cited by 15 publications

References 46 publications

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude

Singular solutions of the BBM equation: analytical and numerical study

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