Symmetry-breaking bifurcation in the nonlinear Schr\"{o}dinger equation with symmetric potentials

TL;DR

This paper analyzes symmetry-breaking bifurcations in the nonlinear Schrödinger equation with symmetric potentials, revealing conditions under which ground states lose symmetry and characterizing the bifurcation types through spectral analysis.

Contribution

It introduces a novel combination of concentration-compactness and spectral analysis techniques to study symmetry-breaking bifurcations in NLS with symmetric potentials, including large eigenvalue regimes.

Findings

Symmetric ground states undergo bifurcation when potential has a non-degenerate maximum.

Bifurcation can be subcritical or supercritical pitchfork depending on parameters.

Large eigenvalue states localize near potential critical points and bifurcate from solitons.

Abstract

We consider the focusing (attractive) nonlinear Schr\"odinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter. We employ a novel technique combining concentration--compactness and spectral properties of linearized Schr\"odinger type operators to show that the symmetric ground states can either be uniquely continued for the entire interval of the eigenvalue parameter or they undergo a symmetry--breaking pitchfork bifurcation due to the second eigenvalue of the linearized operator crossing zero. In addition we prove the appropriate scaling for the stationary states in the limit of large values of the eigenvalue parameter. The scaling and our novel technique imply that all ground states at large eigenvalues must be localized near a critical point of the potential and bifurcate from the soliton of the focusing NLS equation without potential localized at the same point. The theoretical results are illustrated numerically for a double-well potential obtained after the splitting of a single-well potential. We compare the cases before and after the splitting, and numerically investigate bifurcation and stability properties of the ground states which are beyond the reach of our theoretical tools.