The global bifurcation picture for ground states in nonlinear Schrodinger equations

TL;DR

This paper develops a comprehensive method to identify all ground state solutions of nonlinear Schrödinger equations by extending global bifurcation theory and analyzing the structure of solution manifolds.

Contribution

It introduces a novel approach combining global bifurcation theory and local analysis to find all coherent structures supported by nonlinear wave equations.

Findings

All ground states for the nonlinear Schrödinger equation with power nonlinearity are identified.

The method effectively traces solution manifolds to discover coherent structures.

Most structures are found except possibly those forming loops that do not reach the boundary.

Abstract

In this paper, we propose a method of finding all coherent structures supported by a given nonlinear wave equation. It relies on enhancing the recent global bifurcation theory as developed by Dancer, Toland, Buffoni and others, by determining all the limit points of the coherent structure manifolds at the boundary of the Fredholm domain. Local bifurcation theory is then used to trace back these manifolds from their limit points into the interior of the Fredholm domain identifying the singularities along them. This way all coherent structure manifold are discovered except may be the ones which form loops and hence never reach the boundary. The method is then applied to the Schrodinger equation with a power nonlinearity for which all ground states are identified.