Kinks and solitons in the generalized Ginzburg-Landau equation

Abstract

We consider the simplest dynamical model of the Ginzburg-Landau (GL) type with a trivial state that is stable with respect to infinitesimal disturbances but may be triggered into a traveling-wave (TW) state by a finite disturbance. Treating the dispersion coefficients in the GL model as small parameters, we construct a kink solution interpolating between a TW and a trivial state. We find the equilibrium velocity of the kink and demonstrate that it uniquely selects a wave number of the TW. Next we find analytically a stable kink-antikink bound state (a ‘‘soliton’’). In particular, the size of the soliton is found in an explicit form. We also discuss possible implementations of the soliton in particular physical systems.