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

MR2589661 Mathieu COLIN, Thierry COLIN and Masahito OHTA Mathieu COLIN, Thierry COLIN and Masahito OHTA Instability of Standing Waves for a System of Nonlinear Schrödinger Equations with Three-Wave Interaction Funkcialaj Ekvacioj. Serio Internacia 52 2009 371--380 http://fe.math.kobe-u.ac.jp/FE/FullPapers/52-3/52_371.pdf http://www.ams.org/mathscinet-getitem?mr=MR2589661 We consider a three-component system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. In dimension $N\le 3$, we study the orbital instability of standing wave solution of the form $(0,0,e^{2i\omega t}\varphi)$, where $\varphi$ 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. Nonlinear Schrödinger equations, Standing waves, Orbital instability. 35Q55, 35B35. 52-371 2009 Instability of Standing Waves for a System of Nonlinear Schrödinger Equations with Three-Wave Interaction Mathieu COLIN, Thierry COLIN and Masahito OHTA Mathieu COLIN, Thierry COLIN and Masahito OHTA 1 Cazenave, T. Semilinear Schrödinger equations Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI 2003 MR2002047

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