Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

Abstract

For the parametrically driven nonlinear Schrödinger equation, two criteria for the dynamical stability of solitons are presented, using the driving force f(x, t) = a exp[2 i K(t) x]. Both criteria are based on the results of a collective coordinate theory. In this theory, the parameters of the 1-soliton solution are allowed to depend on time and a variational approach yields a set of coupled nonlinear ODEs for the collective coordinates. The solutions of these ODEs are used to compute the soliton’s normalized momentum p(t), velocity v(t), and a parametric plot p(v). The first criterion states that a sufficient condition for instability is fulfilled if the curve p(v), or a piece of it, has a negative slope. If the slope is positive then a necessary condition for stability is fulfilled. For constant K, a second criterion is presented. Here a complex wavefunction is calculated and a parametric plot of its imaginary part vs its real part is made. This yields a closed orbit in the complex plane. If this orbit has a negative sense of rotation with respect to a fixed point, then a sufficient condition for instability is fulfilled, whereas a positive sense is a necessary condition for stability. The ensemble of orbits and fixed points is a so-called phase portrait, which is very helpful for the classification of the different types of solutions of the ODEs. Finally, the validity of the two criteria is tested by comparing with simulations for the driven nonlinear Schrödinger equation. Both criteria make correct predictions about the stability and instability of the solitons, with the exception of a few cases at the border between stability and instability regions.