A solvable model for symmetry-breaking phase transitions

TL;DR

This paper introduces an analytically solvable model for symmetry-breaking phase transitions in solitons pinned to a double-well potential, capturing both first and second kind transitions with explicit solutions.

Contribution

It presents the first solvable model for symmetry-breaking phase transitions of both kinds in solitons within a double-well potential, including explicit solutions.

Findings

Exact solutions for symmetric and asymmetric solitons in the self-focusing case.

Demonstration of switch between sub- and supercritical bifurcations.

Transition of the second kind breaking antisymmetry in the defocusing case.

Abstract

Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.