Instability of Standing Waves for a System of Nonlinear Schrödinger Equations with Three-Wave Interaction

Abstract

We consider a three-component system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. In dimension N ≤ 3, we study the orbital instability of standing wave solution of the form (0,0,e^2iωtψ), where ψ is a ground state of scalar nonlinear Schrödinger equation. Using time derivative instead of space derivatives to estimate nonlinear terms, we improve an instability result in our previous paper [4], and also give a simpler proof.