Standing waves for a Schrödinger system with three waves interaction

Abstract

We study standing waves for a system of nonlinear Schrödinger equations with three waves interaction arising as a model for the Raman amplification in a plasma. We consider the mass-critical and mass-supercritical regimes, and we prove existence of ground states along with a synchronized mass collapse behavior. In addition, we show that the set of ground states is stable under the associated Cauchy flow. Furthermore, in the mass-supercritical setting we construct an excited state that corresponds to a strongly unstable standing wave. Moreover, a semi-trivial limiting behavior of the excited state is drawn accurately. Finally, by a refined control of the excited state's energy, we give sufficient conditions to prove global existence or blow-up of solutions to the corresponding Cauchy problem.

arXiv commentrevised version. slightly shortened, typos fixed

References in corpus (5)

• On dipolar quantum gases in the unstable regime
• Ground States of Two-Component Attractive Bose-Einstein Condensates II: Semi-trivial Limit Behavior
• Orbital stability of standing waves for a system of nonlinear Schrödinger equations with three wave interaction
• Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with ${\chi }^3$ nonlinear response
• Dynamical collapse of cylindrical symmetric Dipolar Bose-Einstein condensates