Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations

TL;DR

This paper develops new techniques for classifying symmetries of nonlinear Schrödinger equations, providing a comprehensive group classification and introducing normalized subclasses to facilitate the analysis of admissible transformations.

Contribution

It introduces the concepts of conditional equivalence groups and normalized classes, and applies these to exhaustively classify symmetries of (1+1)- and (1+2)-dimensional nonlinear Schrödinger equations.

Findings

Complete group classification of (1+1)-dimensional Schrödinger equations with nonlinearities and potentials.

Classification of (1+2)-dimensional cubic Schrödinger equations with potentials.

Development of effective techniques for symmetry analysis of differential equations.

Abstract

The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schroedinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schroedinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1+2)-dimensional cubic Schroedinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.