3 research outputs found
Long time dynamics and coherent states in nonlinear wave equations
We discuss recent progress in finding all coherent states supported by
nonlinear wave equations, their stability and the long time behavior of nearby
solutions.Comment: bases on the authors presentation at 2015 AMMCS-CAIMS Congress, to
appear in Fields Institute Communications: Advances in Applied Mathematics,
Modeling, and Computational Science 201
Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit
We consider the stationary solutions for a class of Schroedinger equations
with a symmetric double-well potential and a nonlinear perturbation. Here, in
the semiclassical limit we prove that the reduction to a finite-mode
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.Comment: 46 pages, 4 figure