Coupled nonlinear Schrödinger equations: spectra and instabilities of plane waves

The coherent coupling of two regular wave trains is considered as a solution of a system of a pair of nonlinear Schrodinger equations with cubic selfand cross-interactions. The purpose is the study of the linear stability of this plane wave solution against small and localised perturbations. The model wave equation considered here is integrable in three different regimes, the focusing, defocusing and mixed regime, according to the values of the coupling constants. The approach we take is based on the construction of the eigenfunctions of the linearised wave equations via the so-called squared eigenfunctions, obtained from the Lax pair. The notion of stability spectrum is introduced as associated in particular to the plane wave solution. This is an algebraic curve in the complex plane of the spectral variable of the Lax pair. By means of the geometric features of the spectrum, we completely classify spectra in the parameter space of amplitudes and coupling constants. The stability is then assessed by computing the eigenfrequencies of the eigenfunctions. The instability band is shown to be related to the properties of the non-real part of the spectrum. This analysis confirms that instabilities exist also in the defocusing regime. Different types of spectra are displayed.