Cuspons, peakons and regular gap solitons between three dispersion curves

A general model is introduced which describes a system with cubic nonlinearities, in a situation when the linear dispersion relation has three branches which nearly intersect. The system includes two waves with a strong linear coupling between them, to which a third wave is coupled. This model has two gaps in its linear spectrum. A nonlinear analysis is performed for zero-velocity solitons. If one may disregard the self-phase modulation (SPM) term in the equation for the third wave, we find an analytical solution which shows that there simultaneously exist two different families of generic solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in the two-wave system, and cuspons, which have finite amplitude and energy, but a singularity in the first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two nearly vanishes, the soliton family remains drastically different from that in the uncoupled system: regular solitons whose amplitude exceeds a critical value are replaced in this limit by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there again simultaneously exist two different families of generic stable soliton solutions, viz., regular ones and peakons, whose existence depends on the sign of certain system parameters. Direct simulations show that both types of the solitons may be stable in this case too. PACS numbers: 05.45.Yv; 42.65.Tg; 42.81.Dp; 47.55.Hd