Symmetry breaking bifurcations in the NLS equation with an asymmetric delta potential

TL;DR

This paper investigates how asymmetry in a double well potential affects symmetry breaking bifurcations in the nonlinear Schrödinger equation, revealing a transition from pitchfork to saddle-centre bifurcations and analyzing solution stability.

Contribution

It provides a rigorous analysis of asymmetry effects on bifurcation types in the NLS equation with double delta potentials, extending understanding beyond symmetric cases.

Findings

Symmetry breaking bifurcation becomes a saddle-centre type with asymmetry.

Standard pitchfork bifurcation is broken due to asymmetry.

Solutions along bifurcation branches are shown to be unstable.

Abstract

We consider the NLS equation with a linear double well potential. Symmetry breaking, i.e., the localisation of an order parameter in one of the potential wells that can occur when the system is symmetric, has been studied extensively. However, when the wells are asymmetric, only a few analytical works have been reported. Using double Dirac delta potentials, we study rigorously the effect of such asymmetry on the bifurcation type. We show that the standard pitchfork bifurcation becomes broken and instead a saddle-centre type is obtained. Using a geometrical approach, we also establish the instability of the corresponding solutions along each branch in the bifurcation diagram