Three wave interaction solitons for an energy critical Schr\"odinger system

Abstract

We investigate standing waves for the energy critical Schr\"odinger system with three waves interaction arising as a model for the Raman amplification in a plasma. Several results are proved: simultaneous existence of stable and unstable standing waves, existence of global solutions, and absence of small data scattering. Our main results show some specific features arising from the three waves interaction differently from the classical energy critical Schr\"odinger equation, and they support some experimental observations on Raman amplification.