Exact solutions of semilinear radial Schrodinger equations by separation of group foliation variables

TL;DR

This paper derives explicit solutions for a class of semilinear radial Schrödinger equations, including new similarity and group-invariant solutions, revealing complex behaviors like blow-up and dispersion, especially for critical nonlinear powers.

Contribution

It introduces a novel separation of group foliation variables method to obtain explicit solutions, surpassing standard symmetry reduction techniques.

Findings

Derived explicit solutions including similarity and non-invariant solutions

Identified solutions with blow-up and dispersion behaviors

Included solutions for critical nonlinear powers

Abstract

Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead to nonlinear ODEs for which few if any explicit solutions can be derived by standard integration methods.