Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit

TL;DR

This paper analyzes stationary solutions of nonlinear Schrödinger equations with double-well potentials in the semiclassical limit, revealing finite-mode approximation validity, symmetry-breaking bifurcations, and stability properties, supported by a simplified delta potential model.

Contribution

It demonstrates that finite-mode approximation accurately captures stationary solutions and bifurcations, and explores stability in nonlinear Schrödinger equations with double-well potentials.

Findings

Finite-mode approximation matches stationary solutions up to exponentially small errors.

Symmetry-breaking bifurcations occur at specific nonlinear strengths.

Stability properties vary across solution branches.

Abstract

We consider the stationary solutions for a class of Schroedinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give the stationary solutions, up to an exponentially small term, and that symmetry-breaking bifurcation occurs at a given value for the strength of the nonlinear term. The kind of bifurcation picture only depends on the non-linearity power. We then discuss the stability/instability properties of each branch of the stationary solutions. Finally, we consider an explicit one-dimensional toy model where the double well potential is given by means of a couple of attractive Dirac's delta pointwise interactions.