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Wednesday, October 14, 2009
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Nonlinear Schrödinger solitons under external spatio-temporal forces and a new stability criterion

Franz Mertens
University of Bayreuth

We investigate the dynamics of solitons of the cubic Nonlinear Schrödinger Equation (NLSE) with the following perturbations: non-parametric spatio-temporal driving of the form f(x,t) = a exp[i K(t) x] and damping. This force has an application in nonlinear optical waveguide arrays.

Using a Lagrangian approach, which is generalized by the introduction of a dissipation function, we develop a Collective-Coordinate-Theory which yields ODEs for the collective coordinates position, velocity, amplitude and phase of the soliton. The ODEs are solved analytically and numerically for the following cases: constant, harmonic and biharmonic K(t). In the first case the spatial average of f(x) vanishes, nevertheless the soliton performs on the average a unidirectional motion. Here the amplitude of oscillations around the average motion is much smaller than the period of f(x). In the biharmonic case the soliton performs a ratchet motion.

We conjecture a new stability criterion: if the curve P(V), where P(t) and V(t) are the soliton momentum and velocity, has a branch with negative slope, the soliton is predicted to become unstable which is confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the shorter the branch with negative slope is, the longer the soliton lives.