On the ground state of the nonlinear Schr{\"o}dinger equation: asymptotic behavior at the endpoint powers

TL;DR

This paper analyzes the asymptotic behavior of ground states of the nonlinear Schrödinger equation at critical nonlinearity powers, establishing convergence to known solutions like Gausson and Aubin-Talenti solitons.

Contribution

It provides rigorous proofs of strong convergence and explicit bounds for ground states at endpoint nonlinearities, along with detailed asymptotics and numerical illustrations.

Findings

Convergence to Gausson at the logarithmic limit

Convergence to Aubin-Talenti soliton in higher dimensions

Explicit bounds and asymptotic descriptions of ground states

Abstract

We consider the ground states of the nonlinear Schr{\"o}dinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity, up to some rescaling to infer non-trivial limits. One case corresponds to the limit towards a Gaussian function called Gausson, which is the ground state of the stationary logarithmic Schr{\"o}dinger equation. The other case, for dimension at least three, corresponds to the limit towards the Aubin-Talenti algebraic soliton. We prove strong convergence with explicit bounds for both cases, and provide detailed asymptotics. These theoretical results are illustrated with numerical approximations.