Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential
Abstract
Here we consider stationary states for nonlinear Schrödinger equations in any spatial dimension n with symmetric double well potentials. These 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 supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddlenode bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; 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 a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n.
 Publication:

Physical Review Letters
 Pub Date:
 November 2009
 DOI:
 10.1103/PhysRevLett.103.194101
 arXiv:
 arXiv:0908.0246
 Bibcode:
 2009PhRvL.103s4101S
 Keywords:

 05.45.a;
 02.30.Oz;
 03.65.Xp;
 03.75.Lm;
 Nonlinear dynamics and chaos;
 Bifurcation theory;
 Tunneling traversal time quantum Zeno dynamics;
 Tunneling Josephson effect BoseEinstein condensates in periodic potentials solitons vortices and topological excitations;
 Mathematical Physics;
 37N20;
 70K50;
 81V55;
 81V70
 EPrint:
 8 pages with 2 figures