Nonlinear Schrodinger equations with multiple-well potential

TL;DR

This paper analyzes stationary solutions of nonlinear Schrödinger equations with multiple-well potentials, revealing how solutions localize and bifurcate, especially in the case of four wells, using semiclassical techniques.

Contribution

It introduces a semiclassical approach to describe ground states in multi-well nonlinear Schrödinger equations, highlighting symmetry-breaking bifurcations and localization phenomena.

Findings

Ground state solutions are described by an N-dimensional Hamiltonian system.

In the four-well case, spontaneous symmetry-breaking bifurcations occur.

For large focusing nonlinearity, solutions localize on a single well.

Abstract

We consider the stationary solutions for a class of Schrodinger equations with a N-well potential and a nonlinear perturbation. By means of semiclassical techniques we prove that the dominant term of the ground state solutions is described by a N-dimensional Hamiltonian system, where the coupling term among the coordinates is a tridiagonal Toeplitz matrix. In particular we consider the case of N=4 wells, where we show the occurrence of spontaneous symmetry-breaking bifurcation effect. In particular, in the limit of large focusing nonlinearity we prove that the ground state stationary solutions consist of N wavefunctions localized on a single well.