Integrability and linear stability of nonlinear waves

TL;DR

This paper introduces a direct, local method to construct eigenmodes for linearized integrable PDEs using their Lax pairs, enabling efficient stability analysis across various systems without relying on spectral data.

Contribution

It develops a general $N imes N$ matrix scheme for constructing eigenmodes directly from Lax pairs, applicable to multi-component integrable equations like coupled nonlinear Schrödinger systems.

Findings

Eigenfrequencies are explicitly computed in the complex spectral plane.

Complete spectral classification in parameter space is provided.

Continuous wave solutions are shown to be generically unstable.

Abstract

It is well known that the linear stability of solutions of partial differential equations which are integrable can be very efficiently investigated by means of spectral methods. We present here a direct construction of the eigenmodes of the linearized equation by using only their associated Lax pair with no reference to spectral data and boundary conditions. This local construction is given in the general $N \times N$ matrix scheme so as to be applicable to a large class of integrable equations, including the multicomponent nonlinear Schroedinger system and the multi-wave resonant interaction system. The analytical and numerical computations involved in this general approach are detailed as an example for $N = 3$ for the particular system of two coupled nonlinear Schroedinger equations in the defocusing, focusing and mixed regimes. The instabilities of the continuous wave solutions are fully discussed in the entire parameter space of their amplitudes and wave numbers. By defining and computing the spectrum in the complex plane of the spectral variable, the eigenfrequencies are explicitly expressed. According to their topological properties, the complete classification of these spectra in the parameter space is presented and graphically displayed. The continuous wave solutions are linearly unstable for a generic choice of the coupling constants.