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

TL;DR

This paper introduces a new integrable long wave-short wave interaction model, analyzes its linear stability spectra through algebraic curves, and classifies the stability of plane wave solutions in parameter space.

Contribution

A novel integrable model generalizing previous long wave-short wave equations, with a comprehensive stability analysis using algebraic geometric methods.

Findings

Complete classification of stability spectra for plane wave solutions.

Identification of modulational instability regions.

Connection between spectral geometry and stability properties.

Abstract

We consider the propagation of short waves which generate waves of much longer (infinite) wave-length. Model equations of such long wave-short wave 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 long wave-short wave integrable model which generalises those first proposed by Yajima-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 linearised long wave-short wave 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.