Standing waves for a Schr\"odinger system with three waves interaction

TL;DR

This paper investigates the existence, stability, and dynamics of standing waves in a three-wave interaction Schrödinger system, with results on ground states, excited states, and solution behaviors.

Contribution

It establishes the existence and stability of ground states, constructs an unstable excited state, and provides conditions for global existence or blow-up in the system.

Findings

Existence of ground states in mass-critical and supercritical regimes.

Stability of ground states under the Cauchy flow.

Construction of an unstable excited state with semi-trivial limiting behavior.

Abstract

We study standing waves for a system of nonlinear Schr\"odinger 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.