Modulational Instability in Equations of KdV Type

Abstract

It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristicsamplitude, phase, wave number, etc. -slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understanding of the dynamics, in particular, the instability of slowly modulated wave trains for nonlinear dispersive equations of KdV type.

Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

“…By Floquet theory, we know the spectrum associated with is purely essential, containing no isolated eigenvalues of finite multiplicity: see . In particular, see [, proposition 3.1] for Floquet theory that applies to nonlocal operators.…”

Modulational instability in nonlinear nonlocal equations of regularized long wave type

“…Since the spectrum of L is symmetric with respect to the reflections about the real and imaginary axes, u is spectrally unstable if and only if the L 2 (R)-spectrum of L is not contained in the imaginary axis. It follows from Floquet theory (see [BHJ16], for instance, and references therein) that nontrivial solutions of (2.7) cannot be integrable over R. Rather they are at best bounded over R. Furthermore it is well-known that the L 2 (R)-spectrum of L is essential. In the case of the KdV equation, for instance, the (essential) spectrum of the associated linearized operator may be studied with the help of Evans function techniques and other ODE methods.…”

“…Confronted with a nonlocal operator, unfortunately, they are not viable to use. Instead, it follows from Floquet theory (see [BHJ16], for instance, and references therein) that λ ∈ C belongs to the L 2 (R)-spectrum of L if and only if…”

“…We repeat the argument in the previous section (see also [HJ15a,BHJ16] and references therein) to learn that λ ∈ C belongs to the L 2 (R) × L 2 (R)-spectrum of L if and only if…”

“…Since then, there has been considerable interest in the mathematical community in rigorously justifying predictions from Whitham's modulation theory. Recently in [BH14,Joh13,HJ15a,HJ15b] (see also [BHJ16]), in particular, long wavelength perturbations were carried out analytically for (1.7) and for a class of Hamiltonian systems permitting nonlocal dispersion, for which Evans function techniques and other ODE methods may not be applicable. Specifically, modulational instability indices were derived either with the help of variational structure (see [BH14]) or using asymptotic expansions of the solution (see [Joh13,HJ15a,HJ15b]).…”