Solitary Waves under the Influence of a Long-Wave Mode

The dynamics of solitons of the nonlinear Schr\"odinger equation under the influence of dissipative and dispersive perturbations is investigated. In particular a coupling to a long-wave mode is considered using extended Ginzburg-Landau equations. The study is motivated by the experimental observation of localized wave trains (`pulses') in binary-liquid convection. These pulses have been found to drift exceedingly slowly. The perturbation analysis reveals two distinct mechanisms which can lead to a `trapping' of the pulses by the long-wave concentration mode. The are given by the effect of the concentration mode on the local growth rate and on the frequency of the wave. The latter, dispersive mechanism has not been recognized previously, despite the fact that it dominates over the dissipative contribution within the perturbation theory. A second unexpected result is that the pulse can be accelerated by the concentration mode despite the reduced growth rate ahead of the pulse. The dependence of the pulse velocity on the Rayleigh number is discussed, and the hysteretic `trapping' transitions suggested by the perturbation theory are confirmed by numerical simulations, which also reveal oscillatory behavior of the pulse velocity in the vicinity of the transition. The derivation and reconstitution of the extended Ginzburg-Landau equations is discussed in detail.