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 [LBJM2019], 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 [LBJM2019] 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. This paper deals with modulational stability of equation ( 1), that is, the spectral stability subject to long-wavelength perturbations. Rigorously speaking, this amounts to expanding the spectrum of the linearized operator in a neighbourhood of the origin in the spectral plane. Whitham modulation theory [

About plane periodic waves of the nonlinear Schrödinger equations

“…x , ρ p0q , k p0q φ q ă 0. Again, these results are new in this context, except for the corollary about weak hyperbolicity that overlaps with the recent preprint [CM20] -based on the recent [LBJM21] -, appeared during the preparation of the present contribution. Note however that our proof of the corollary is different and our assumptions are considerably weaker.…”

“…In particular no connection with geometrical optics and modulated systems is offered for the matrix r Λ 0 . This connection is established in a recent preprint [CM20], building upon [LBJM21]. Hence the next remarks also apply to [CM20].…”

“…This connection is established in a recent preprint [CM20], building upon [LBJM21]. Hence the next remarks also apply to [CM20]. (c) The structure of eigenfunctions is left out of the discussion in [LBJM21], whereas this is our main motivation for reproving in a different way the second part of Corollary 4.5.…”

“…17 Incidentally we point out that from the homogeneity of first-order systems it follows that ill-posedness and stability are essentially the same for systems such as (4.8). 18 Except for the almost simultaneous [CM20]. See detailed comparison in Section 4.3.…”