Exponential decay of the solutions to nonlinear Schr\"odinger systems

TL;DR

This paper proves that solutions to general nonlinear Schrödinger systems decay exponentially at infinity, regardless of their sign or interaction type, and uses this to bound the minimal energy for solutions with specific component configurations.

Contribution

It establishes exponential decay for solutions to a broad class of nonlinear Schrödinger systems, including sign-changing and mixed-interaction cases, and derives energy bounds based on decay properties.

Findings

Solutions decay exponentially at infinity

Applicable to positive, sign-changing, cooperative, and competitive systems

Provides bounds for minimal energy with prescribed component signs

Abstract

We show that the components of finite energy solutions to general nonlinear Schr\"odinger systems have exponential decay at infinity. Our results apply to positive or sign-changing components, and to cooperative, competitive, or mixed-interaction systems. As an application, we use the exponential decay to derive an upper bound for the least possible energy of a solution with a prescribed number of positive and nonradial sign-changing components.