Bifurcation and Stability for Nonlinear Schrödinger Equations with Double Well Potential in the Semiclassical Limit

Abstract

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 finitemode 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.

Stability and Symmetry-Breaking Bifurcation for the Ground States of a NLS with a δ′ Interaction

“…Second, we claim that 18) and such a value is attained at t 1 = t 2 =t. To show this, we use the Lagrange multiplier method, and find that any stationary point of the function t −1…”

“…Nevertheless, a situation similar in some respects appears in the case of the NLS with two attractive δ interactions separated by a distance, a double δ -well. This model is studied in [26], where an analysis of the model is given by means of dynamical systems techniques, and [18], where the bifurcation is explored in the semiclassical regime. See also [27] for the analogous phenomenon with a regular potential of double well type and [30] for the introduction of a related normal form.…”

Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation

“…Based on the two-mode reduction, Sacchetti [15] also reported the same threshold p * as in (1.9) that separates the supercritical and subcritical pitchfork bifurcations. Recently, Fukuizumi and Sacchetti [3] justified the two-mode reduction rigorously in the semi-classical limit, up to an exponentially small error term.…”

Nonlinear Stark--Wannier Equation

“…Remark 5. The standard "tight binding" model is obtained by substituting (12) in (3), and it reduces (3) to a discrete nonlinear Schrödinger equation. In fact, in order to improve the estimate of the remainder terms of the discrete nonlinear Schrödinger equation we decompose the wave function ψ(x) on a different base where the vectors of such a base are obtained by means of the single well semiclassical approximation described in §3.2.…”