Lie symmetries of semi-linear Schr\"odinger equations and applications

TL;DR

This paper explores the conditional Lie symmetries of semi-linear 1D Schr"odinger and diffusion equations, revealing their connection to conformal Lie algebras and classifying invariant equations with applications in physics.

Contribution

It introduces a novel approach by considering mass as an additional variable, linking symmetries to conformal Lie algebra subalgebras, and provides a complete classification of conditionally invariant equations.

Findings

Classified representations of parabolic subalgebras of conf_3.

Derived the complete list of conditionally invariant semi-linear Schr"odinger equations.

Discussed applications to phase-ordering kinetics and particle-reaction models.

Abstract

Conditional Lie symmetries of semi-linear 1D Schr\"odinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schr\"odinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra conf_3. The corresponding representations of the parabolic and almost-parabolic subalgebras of conf_3 are classified and the complete list of conditionally invariant semi-linear Schr\"odinger equations is obtained. Applications to the phase-ordering kinetics of simple magnets and to simple particle-reaction models are briefly discussed.