applied math seminar
Title: Stability criteria for solitons of the nonlinear Schrödinger Equation with spatio-temporal forcing
As an introduction the main properties of solitons are reviewed. Several examples are presented, together with some applications.
We investigate the dynamics of traveling oscillating solitons of the nonlinear Schroedinger (NLS) equation under an external spatio-temporal forcing of the form f (x,t ) = a exp[iK(t )x]. We conjecture a stability criterion for these solitons which is based on a collective coordinate theory. The criterion predicts that the soliton will be unstable if the “stability curve” p(v), where p(t) and v(t ) are the normalized momentum and the velocity of the soliton, has a section with a negative slope. This is confirmed by simulations for the NLS equation.
In the case of a constant K and zero damping, we use the collective coordinate solutions to compute a “phase portrait” of the soliton where its dynamics is represented by an orbit in a phase space. We conjecture, and confirm by simulations, that the soliton is unstable if the orbit has a part with a negative sense of rotation.
For the case of the NLS equation with a complex potential we consider the following cases: trapped or traveling oscillating solitons, quasi-periodic oscillations, blow-up or vanishing solitons. All results are confirmed by simulations.
Contact Name: Pavel Lushnikov