Modulational Instability and Variational Structure

Abstract

We study the modulational instability of periodic traveling waves for a class of Hamiltonian systems in one spatial dimension. We examine how the Jordan block structure of the associated linearized operator bifurcates for small values of the Floquet exponent to derive a criterion governing instability to long wavelengths perturbations in terms of the kinetic and potential energies, the momentum, the mass of the underlying wave, and their derivatives. The dispersion operator of the equation is allowed to be nonlocal, for which Evans function techniques may not be applicable. We illustrate the results by discussing analytically and numerically equations of Korteweg-de Vries type. 1 The requirement that the equation is in one spatial dimension merely enters in the discussion of a Pohozaev-type identity, which is to avoid inversion of a linearized operator; see Lemma 9. 2 Note, however, that the approach in [51,52], for instance, realizing the Evans function as a regularized Fredholm determinant may be more generally applicable than the standard ODE-based formulation.

Modulational instabilities of periodic traveling waves in deep water

“…Although much of the asymptotic work regarding Benjamin-Feir dates back to the 1960s and RIT, more recently a number of authors have been pursuing rigorous proof of the existence of this instability in a variety of wave models [37,38]. Most similar in spirit to this work is that in [39,40], where an analogous perturbation in Bloch parameter is used. Although we consider only formal asymptotics, such asymptotics have been used as the basis for proofs of the existence of solutions in the TFE framework [31,30].…”

Modulational instability in a full‐dispersion shallow water model

“…Hence, they are not directly applicable to (3) or (4), and (2), which involve a nonlocal operator. The authors and collaborators 11,[20][21][22] (see also Ref. 19) instead worked out the corresponding long wavelength perturbation in a rigorous manner, whereby they successfully determined modulational stability and instability for a broad class of nonlinear dispersive equations, permitting nonlocal operators.…”

“…Hence, we merely hit the main points. We use (20) whenever it is convenient to do so. Lemma 5.1 (Regularity).…”

Modulational instability in nonlinear nonlocal equations of regularized long wave type

“…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]).…”