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

For the parametrically driven nonlinear Schrodinger equation, two criteria for the dynamical stability of solitons are presented, using the driving forcef(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 momentump(t), velocityv(t), and a parametric plotp(v). The first criterion states that asufficientcondition for instability is fulfilled if the curvep(v), or a piece of it, has a negative slope. If the slope is positive then anecessarycondition for stability is fulfilled. For constantK, 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 asufficientcondition for instability is fulfilled, whereas a positive sense is anecessarycondition for stability. The ensemble of orbits and fixed points is a so-calledphase 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 Schrodinger 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.