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 family is characterized by two independent parameters, a soliton potential and a soliton velocity. It is shown that this system, in the general situation, is not Galilean invariant. As a result, the family of movable solitons cannot be obtained from the rest soliton solution by applying the corresponding Galilean transformation. The region of soliton parameters is found analytically and confirmed by numerical integration of the steady equations. On the boundary of the region, the solitons bifurcate. For this system, there exist two kinds of bifurcation: supercritical and subcritical. In the first case, the soliton amplitudes vanish smoothly as the boundary is approached. Near the bifurcation point the soliton form is universal, determined from the nonlinear Schrödinger equation. For the second type of bifurcation the wave amplitudes remain finite at the boundary. In this case, the Manley–Rowe integral increases indefinitely as the boundary is approached, and therefore according to the VK-type stability criterion, the solitons are unstable.