Symmetry breaking in competing single-well linear-nonlinear potentials

TL;DR

This paper investigates how competing linear and nonlinear single-well potentials induce spontaneous symmetry breaking in the ground state of the nonlinear Schrödinger equation, revealing different bifurcation scenarios depending on the nonlinearity profile.

Contribution

It introduces a novel analysis of symmetry breaking caused by competition between linear and nonlinear potentials with different profiles, supported by analytical and numerical results.

Findings

Symmetry breaking occurs at a critical norm value.

Bounded nonlinear profiles lead to a pitchfork bifurcation.

Unbounded nonlinear profiles cause an inverted bifurcation.

Abstract

The combination of linear and nonlinear potentials, both shaped as a single well, enables competition between the confinement and expulsion induced by the former and latter potentials, respectively. We demonstrate that this setting leads to spontaneous symmetry breaking (SSB) of the ground state in the respective generalized nonlinear Schr\"{o}dinger (Gross-Pitaevskii) equation, through a spontaneous off-center shift of the trapped mode. Two different SSB bifurcation scenarios are possible, depending on the shape of the nonlinearity-modulation profile, which determines the nonlinear potential. If the profile is bounded (remaining finite at $|x|\rightarrow \infty $), at a critical value of the integral norm the spatially symmetric state loses its stability, giving rise to a pair of mutually symmetric stable asymmetric ones via a direct pitchfork bifurcation. On the other hand, if the nonlinear potential is unbounded, two unstable asymmetric modes merge into the symmetric metastable one and destabilize it, via an inverted pitchfork bifurcation. Parallel to a systematic numerical investigation, basic results are obtained in an analytical form. The settings can be realized in Bose-Einstein condensates and nonlinear optical waveguides.