Rigorous justification of the Whitham modulation theory for equations of NLS type

Abstract

We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi‐linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from Stability of Travelling wave solutions of Nonlinear Dispersive equations of NLS type, where Leisman et al. compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of Leismen et al.​to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave.

Solitons on the rarefaction wave background via the Darboux transformation

“…One of the important problems is to use the Darboux transformation for computations of the limiting phase shifts of the transmitted solitary waves as tfalse→±normal∞ and comparison with the experimentally detected phase shifts [4]. Another interesting problem is to understand better how the modulation theory for soliton propagation used in our data analysis is justified within the Whitham modulation theory [9–11]. Although the resolution formulas for N solitons transmitted over the zero background have been derived in refs.…”

“…A dual problem was the interaction of a linear wavepacket (modulated waves) with the step-like initial data [8]. Both transmission and trapping conditions of a small amplitude, linear, dispersive wave propagating through an expansion (RW) or a undular bore (DSW) were explained by using the Whitham modulation theory [9][10][11].…”

Rigorous justification of the Whitham modulation equations for equations of Whitham type

“…To justify the assumption that we may neglect all the convolution terms involving remainders at O(ϵ), we note that the formal procedure for determining modulational instability from the first-order Whitham modulation equations involves linearizing the equations around a constant solution. 2,13,14,16,17 We may restrict the class of functions 𝑢(𝜃, 𝑋, 𝑇) so that 𝑢 is bounded in 𝑋, 𝑇 to derive the modulation equations to first order. However, the following lemma outlines a weaker set of assumptions which also yield the desired vanishing of remainder terms at O(𝜖).…”

“…There is, however, a growing body of evidence that general dispersive PDEs allow these two theories to coincide. In particular, proofs exist for: systems of viscous conservation laws 11,12 ; the generalized KdV equation 13 ; the nonlinear Klein–Gordon equation 14 ; systems of nondissipative, local Hamiltonian equations 15 ; a viscous fluid conduit equation 16 ; and the generalized nonlinear Schrödinger equation 17 . The primary goal of this paper is to present the first proof of the agreement between Whitham modulation theory and spectral stability to long‐wavelength perturbations for the generalized (nonlocal) Whitham equation (), a primary benefit of which is the concise justification of Whitham modulation theory for all equations that can be expressed in the form Equation () satisfying the following assumptions.…”