On scalar-type standing-wave solutions to systems of nonlinear Schr\"odinger equations

TL;DR

This paper investigates scalar-type standing-wave solutions in systems of nonlinear Schrödinger equations, providing conditions for their existence, characterization, and stability, with a broad abstract framework applicable to various NLS systems.

Contribution

It offers a comprehensive analysis of scalar-type ground and excited states in NLS systems, including existence, shape, and stability criteria, using a novel abstract approach.

Findings

Necessary and sufficient conditions for ground state existence.

Characterization of ground states as scalar multiples of a fixed function.

Conditions for the existence of excited states and their properties.

Abstract

In this article, we study the standing-wave solutions to a class of systems of nonlinear Schr\"odinger equations. Our target is all the standard forms of the NLS systems, with two unknowns, that have a common linear part and cubic gauge-invariant nonlinearities and that yield a Hamiltonian with a coercive kinetic-energy part. We give a necessary and sufficient condition on the existence of the ground state. Further, we give a characterization of the shape of the ground state. It will turn out that the ground states are scalar-type, i.e., multiples of a constant vector and a scalar function. We further give a sufficient condition on the existence of excited states of the same form. The stability and the instability of the ground states are also studied. To this end, we introduce an abstract treatment on the study of scalar-type standing-wave solution that applies to a wide class of NLS systems with homogeneous energy-subcritical nonlinearity. By the argument, some previous results are reproduced.