Effects of Long-Range Nonlinear Interactions in Double-Well Potentials

TL;DR

This paper investigates how long-range nonlinear interactions influence the symmetry-breaking bifurcations in double-well potentials, revealing that the bifurcation structure remains but the critical points are affected by the interaction range.

Contribution

It introduces a detailed analysis of the effects of nonlocal interactions on SSB bifurcations in DWP systems, including a new few-mode approximation and bifurcation analysis.

Findings

SSB bifurcation structure persists with long-range interactions.

Critical points of SSB are sensitive to interaction range.

Complex bifurcation structures, including subcritical bifurcations, are identified.

Abstract

We consider the interplay of linear double-well-potential (DWP) structures and nonlinear longrange interactions of different types, motivated by applications to nonlinear optics and matter waves. We find that, while the basic spontaneous-symmetry-breaking (SSB) bifurcation structure in the DWP persists in the presence of the long-range interactions, the critical points at which the SSB emerges are sensitive to the range of the nonlocal interaction. We quantify the dynamics by developing a few-mode approximation corresponding to the DWP structure, and analyze the resulting system of ordinary differential equations and its bifurcations in detail. We compare results of this analysis with those produced by the full partial differential equation, finding good agreement between the two approaches. Effects of the competition between the local self-attraction and nonlocal repulsion on the SSB are studied too. A far more complex bifurcation structure involving the possibility for not only supercritical but also subcritical bifurcations and even bifurcation loops is identified in that case.