Mixed type (bright-dark) soliton solutions of the integrable N-coupled nonlinear Schrödinger (CNLS) equations with mixed signs of focusing and defocusing type nonlinearity coefficients are obtained by using Hirota's bilinearization method. Generally, for the mixed N-CNLS equations the bright and dark solitons can be split up in (N − 1) ways. By analysing the collision dynamics of these coupled bright and dark solitons systematically we point out that for N > 2, if the bright solitons appear in at least two components, non-trivial effects like onset of intensity redistribution, amplitude dependent phase-shift and change in relative separation distance take place in the bright solitons during collision. However their counterparts, the dark solitons, undergo elastic collision but experience the same amplitude dependent phase-shift as that of bright solitons. Thus in the mixed CNLS system there co-exist shape changing collision of bright solitons and elastic collision of dark solitons with amplitude dependent phase-shift, thereby influencing each other mutually in an intricate way.

Abstract. We obtain explicit bright one-and two-soliton solutions of the integrable case of the coherently coupled nonlinear Schrödinger equations by applying a nonstandard form of the Hirota's direct method. We find that the system admits both degenerate and non-degenerate solitons in which the latter can take single hump, double hump, and flat-top profiles. Our study on the collision dynamics of solitons in the integrable case shows that the collision among degenerate solitons and also the collision of non-degenerate solitons are always standard elastic collisions. But the collision of a degenerate soliton with a non-degenerate soliton induces switching in the latter leaving the former unaffected after collision, thereby showing a different mechanism from that of the Manakov system.

Abstract. Bright plane soliton solutions of an integrable (2+1) dimensional (n + 1)-wave system are obtained by applying Hirota's bilinearization method. First, the soliton solutions of a 3-wave system consisting of two short wave components and one long wave component are found and then the results are generalized to the corresponding integrable (n + 1)-wave system with n short waves and single long wave. It is shown that the solitons in the short wave components (say S(1) and S (2) ) can be amplified by merely reducing the pulse width of the long wave component (say L). The study on the collision dynamics reveals the interesting behaviour that the solitons which split up in the short wave components undergo shape changing collisions with intensity redistribution and amplitude-dependent phase shifts. Even though similar type of collision is possible in (1+1) dimensional multicomponent integrable systems, to our knowledge for the first time we report this kind of collisions in (2+1) dimensions. However, solitons which appear in the long wave component exhibit only elastic collision though they undergo amplitude-dependent phase shifts.

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We derive a (2+1)-dimensional multicomponent long-wave-short-wave resonance interaction (LSRI) system as the evolution equation for propagation of N-dispersive waves in weak Kerr-type nonlinear medium in the small-amplitude limit. The mixed- (bright-dark) type soliton solutions of a particular (2+1)-dimensional multicomponent LSRI system, deduced from the general multicomponent higher-dimensional LSRI system, are obtained by applying the Hirota's bilinearization method. Particularly, we show that the solitons in the LSRI system with two short-wave components behave like scalar solitons. We point out that for an N-component LSRI system with N>3, if the bright solitons appear in at least two components, interesting collision behavior takes place, resulting in energy exchange among the bright solitons. However, the dark solitons undergo standard elastic collision accompanied by a position shift and a phase shift. Our analysis on the mixed bound solitons shows that the additional degree of freedom which arises due to the higher-dimensional nature of the system results in a wide range of parameters for which the soliton collision can take place.