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

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

2011-04-08

arxiv.org/abs/1104.1511v1

Mathematics

Analysis of PDEs

46 pages, 4 figures

Scientific paper

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.

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