A new integrable model of long wave-short wave interaction and linear stability spectra.

Marcos Caso-Huerta,Antonio Degasperis,Sara Lombardo,Matteo Sommacal

doi:10.1098/rspa.2021.0408

Abstract

We consider the propagation of short waves which generate waves of much longer (infinite) wavelength. Model equations of such long wave–short wave (LS) resonant interaction, including integrable ones, are well known and have received much attention because of their appearance in various physical contexts, particularly fluid dynamics and plasma physics. Here we introduce a new LS integrable model which generalizes those first proposed by Yajima and Oikawa and by Newell. By means of its associated Lax pair, we carry out the linear stability analysis of its continuous wave solutions by introducing the stability spectrum as an algebraic curve in the complex plane. This is done starting from the construction of the eigenfunctions of the linearized LS model equations. The geometrical features of this spectrum are related to the stability/instability properties of the solution under scrutiny. Stability spectra for the plane wave solutions are fully classified in the parameter space together with types of modulational instabilities.

Highlights

We consider the propagation of short waves which generate waves of much longer wavelength
The change of sign makes the factor Q21(ζ ), see (3.31b), irrelevant to this analysis, so that hereafter we focus our attention on the factor Q2(ζ ) only
We exploited integrability to construct a method to predict whether a nonlinear wave, described by an integrable, nonlinear system, is linearly stable against small perturbations

Summary

Integrable models of long wave–short wave resonant interaction

Waves are represented as solutions of nonlinear partial differential equations. Where ξ = ε(x − vt) and t2 = ε2t are the rescaled space and time coordinates, and the asterisk indicates complex conjugation This equation follows via the multiscale method from almost any real propagation equation and provides the lowest-order effect of the nonlinear terms on the linear solution εS(ξ , t2) ei(kx−ωt) + εS∗(ξ , t2) e−i(kx−ωt), t2 = ε2t, ξ = ε(x − vt), given by the sum of two plane waves with amplitudes S and S∗, and with wavenumber k, frequency ω and group velocity v. The integrability method has been used to unveil the stability properties of plane wave solutions of two coupled NLS equations [16,21] For these model equations, instabilities have been fully classified in terms of coupling constants, amplitudes and wavenumbers, including instability effects due to defocusing self- and cross-interactions.

Integrable equations and small variation of their solutions

Interaction of long and short continuous waves

Summary and conclusion