We consider the stationary solutions for a class of Schrödinger 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 nonlinearity 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.
- Nonlinear Schrödinger equation
- Orbital stability
- Spontaneous symmetry breaking bifurcation
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics