On asymptotic nonlocal symmetry of nonlinear Schr\"odinger equations

TL;DR

This paper introduces the concept of asymptotic symmetry for nonlinear Schrödinger equations, extending known nonlocal invariances to a broader class of equations and identifying solutions with enhanced smoothing properties.

Contribution

It defines asymptotic symmetry based on reducibility relative to an invariant ansatz and extends nonlocal Lorentz invariance to nonlinear Schrödinger equations.

Findings

Extension of nonlocal Lorentz invariance to nonlinear equations

Identification of solutions with improved smoothing properties

Introduction of asymptotic symmetry concept

Abstract

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schr\"odinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schr\"odinger equations. An important class of solutions of the free Schr\"odinger equation with improved smoothing properties is obtained.