Théorie de la diffusion quantique pour des perturbations à longue et courte portée du champ magnétique
This text is a survey of recent results on traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. We present the existence, nonexistence and stability results and we describe the main ideas used in proofs.
We study vortices for solutions of the perturbed Ginzburg–Landau equations where is estimated in . We prove upper bounds for the Ginzburg–Landau energy in terms of , and obtain lower bounds for in terms of the vortices when these form “unbalanced clusters” where . These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow, including...
We deduce from the first part of this paper [S1] estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow us to study various phenomena occurring in this flow, including vortex collisions; they allow in particular extending the dynamical law of vortices past collision times.