Asymptotics of an autoresonance soliton

Abstract

Phase locking is studied in a one-dimensional medium under the action of an external force with slowly changing frequency. In a typical situation, the phase locking is described by a nonstationary nonlinear Schr¨odinger equation with external force. For large values of the time variable, the leading term of a space-localized growing asymptotic solution with soliton profile in the main order is constructed. It turned out that a time-growing asymptotic solution can be obtained for an external excitation with decreasing amplitude. Necessary growth conditions are deduced for such a solution in the presence of dissipation.