On Existence of Solitons for the 3rd Harmonic of a Light Beam in Planar Waveguides

We consider equations governing propagation of a monochromatic laser beam coupled to its third harmonic in a nonlinear medium. The system proves to be non-integrable in the sense of Painleve. However it is partially integrable for all values of its parameters. We further check the possibility of solving the equations by the Hirota bilinear method. The system is found to be solvable this way provided that the complex phase of the third harmonics is equal to tripled phase of the fundamental mode (modulo i) and also the amplitudes of these modes are in special proportion. This result corresponds to the previously known condition of existence of the sech soliton solutions. Furthermore, the Hirota scheme is found to work only for exact resonance, i.e. for the ratio of the dispersion coefficients equal to the ratio of frequencies. Finally, all these conditions may only be satisfied for single envelope solitons of the cubic Schrödinger type.