Dynamical symmetries of semi-linear Schr\"odinger and diffusion equations

TL;DR

This paper explores the symmetries of semi-linear Schr"odinger and diffusion equations by extending the variables to include mass or diffusion constant, revealing connections to conformal Lie algebras and classifying invariant equations.

Contribution

It introduces a novel approach by considering mass as an additional variable, linking symmetries to conformal Lie algebras, and classifies all conditionally invariant semi-linear Schr"odinger equations.

Findings

Classification of non-hermitian representations

Complete list of conditionally invariant equations

Insights into dynamical scaling in phase-ordering kinetics

Abstract

Conditional and 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. We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. 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. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed.