Group-invariant solutions of semilinear Schrodinger equations in multi- dimensions

TL;DR

This paper systematically derives all explicit radial solutions invariant under symmetry groups for a class of semilinear Schrödinger equations in multiple dimensions, including critical cases, using symmetry reduction and conservation laws.

Contribution

It provides a comprehensive method to find explicit group-invariant solutions for semilinear Schrödinger equations in multiple dimensions, including critical and non-critical cases.

Findings

All explicit group-invariant radial solutions are obtained.

Conservation laws associated with symmetry groups are fully listed.

The approach applies to both focusing and defocusing nonlinearities.

Abstract

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schrodinger equations in dimensions $n\neq 1$. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the conformal power $p=4/n$ relevant for critical dynamics.The methods involve, firstly, reduction of the semilinear Schrodinger equations to group-invariant complex 2nd order ODEs with respect to an optimal set of one-dimensional point symmetry groups, and secondly, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether's theorem, all conservation laws arising from these point symmetry groups are listed.