The two-parameter soliton family for the interaction of a fundamental and its second harmonic

For a system of interacting fundamental and second harmonics, the soliton f amily is characterized by two independent parameters, a soliton potential a nd a soliton velocity. It is shown that this system, in the general situati on, is not Galilean invariant. As a result, the family of movable solitons cannot be obtained from the rest soliton solution by applying the correspon ding Galilean transformation. The region of soliton parameters is found ana lytically and confirmed by numerical integration of the steady equations. O n the boundary of the region, the solitons bifurcate. For this system, ther e exist two kinds of bifurcation: supercritical and subcritical. In the fir st case, the soliton amplitudes vanish smoothly as the boundary is approach ed. Near the bifurcation point the soliton form is universal, determined fr om the nonlinear Schrodinger equation. For the second type of bifurcation t he wave amplitudes remain finite at the boundary. In this case, the Manley- Rowe integral increases indefinitely as the boundary is approached, and the refore according to the VK-type stability criterion, the solitons are unsta ble. (C) 2001 Elsevier Science B.V. All rights reserved.