Transformation from the nonautonomous to standard NLS equations

TL;DR

This paper introduces a systematic transformation method to derive exact solutions of the nonautonomous nonlinear Schrödinger equation from the well-studied standard NLS solutions, facilitating control over soliton dynamics.

Contribution

It presents a new transformation approach based on integrability conditions that links solutions of the standard NLS to the nonautonomous version, expanding solution techniques.

Findings

Derived a general transformation for nonautonomous NLS solutions

Validated the method with bright and dark soliton examples

Provided a way to control soliton dynamics explicitly

Abstract

In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schr\"odinger (NLS) equation. An integrable condition is first obtained by the Painlev\`e analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.