Universal critical power for nonlinear Schrodinger equations with symmetric double well potential

TL;DR

This paper investigates bifurcation phenomena in nonlinear Schrödinger equations with symmetric double well potentials, revealing a universal critical nonlinearity power that determines the bifurcation type in the semiclassical limit.

Contribution

It identifies a universal critical nonlinearity power value (2) that governs bifurcation behavior in these equations, independent of potential shape.

Findings

For nonlinearity power less than 2, a simple pitch-fork bifurcation occurs.

For nonlinearity power greater than 2, a saddle point collapse leads to inverse pitch-fork bifurcation.

The critical value of 2 is universal, not depending on potential shape.

Abstract

Here we consider stationary states for nonlinear Schrodinger equations with symmetric double well potentials. These stationary states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a simple pitch-fork bifurcation, and a couple of saddle points which unstable branches collapse in an inverse pitch-fork bifurcation. In this paper we show that in the semiclassical limit, or when the barrier between the two wells is large enough, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value (2); in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is an universal constant in the sense that it does not depends on the shape of the double well potential.