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

TL;DR

This paper develops a normal form reduction for the nonlinear Schrödinger equation to analyze symmetry-breaking bifurcations in double-well potentials, proving the persistence of bifurcation dynamics over finite times.

Contribution

It introduces a rigorous normal form reduction for pitchfork bifurcations in the nonlinear Schrödinger equation with symmetric potentials, applicable to both supercritical and subcritical cases.

Findings

Normal form reduction accurately captures bifurcation dynamics.

Persistence of bifurcation behavior over long finite times.

Applicable to general symmetric double-well potentials.

Abstract

We derive and justify a normal form reduction of the nonlinear Schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time-dependent solutions of the nonlinear Schrodinger equation over long but finite time intervals.