We present the exact bright one-soliton and two-soliton solutions of the integrable three coupled nonlinear Schrödinger equations (3-CNLS) by using the Hirota method, and then obtain them for the general N -coupled nonlinear Schrödinger equations (N -CNLS). It is pointed out that the underlying solitons undergo inelastic (shape changing) collisions due to intensity redistribution among the modes. We also analyse the various possibilities and conditions for such collisions to occur. Further, we report the significant fact that the various partial coherent solitons (PCS) discussed in the literature are special cases of the higher order bright soliton solutions of the N -CNLS equations.PACS numbers: 42.81. Dp, 42.65.Tg In recent years the concept of soliton has been receiving considerable attention in optical communications since soliton is capable of propagating over long distances without change of shape and with negligible attenuation [1][2][3]. It has been found that soliton propagation through optical fiber arrays is governed by a set of equations related to the CNLS equations [1,2],where q j is the envelope in the jth core, z and t represent the normalized distance along the fiber and the retarded time, respectively. Here 2µ gives the strength of the nonlinearity. Eq. (1) reduces to the standard envelope soliton possessing integrable nonlinear Schrödinger equation for N = 1. For N = 2, the above Eq. (1) governs the integrable Manakov system [4] and recently for this case the exact two-soliton solution has been obtained and novel shape changing inelastic collision property has been brought out [5]. However, the results are scarce for N ≥ 3, even though the underlying systems are of considerable physical interest. For example, in addition to optical communication, in the context of biophysics the case N = 3 can be used to study the launching and propagation of solitons along the three spines of an alpha-helix in protein [6]. Similarly the CNLS Eq.(1) and its generalizations for N ≥3 are of physical relevance in the theory of soliton wavelength division multiplexing [7], multi-channel bit-parallel-wavelength optical fiber networks [8] and so on. In particular, for arbitrary N , Eq.(1) governs the propagation of N -self trapped mutually incoherent wavepackets in Kerr-like photorefractive media [9] in which q j is the jth component of the beam, z and t represents the normalized coordinates along the direction of propagation and the transverse coordinate, respectively, and N p=1 |q p | 2 represents the change in the refractive index profile created by all incoherent components of the light beam [9] when the medium response is slow.The parameter µ = k 2 0 n 2 /2, where n 2 is the nonlinear Kerr coefficient and k 0 is the free space wave vector.In this letter, we report the exact bright one and two soliton solutions, first for the N = 3 case and then for the arbitrary N case, where the procedure can be extended in principle to higher order soliton solutions, using the Hirota bilinearization method. In particular, ...

The different dynamical features underlying soliton interactions in coupled nonlinear Schrödinger equations, which model multimode wave propagation under varied physical situations in nonlinear optics, are studied. In this paper, by explicitly constructing multisoliton solutions (up to four-soliton solutions) for two-coupled and arbitrary N-coupled nonlinear Schrödinger equations using the Hirota bilinearization method, we bring out clearly the various features underlying the fascinating shape changing (intensity redistribution) collisions of solitons, including changes in amplitudes, phases and relative separation distances, and the very many possibilities of energy redistributions among the modes of solitons. However, in this multisoliton collision process the pairwise collision nature is shown to be preserved in spite of the changes in the amplitudes and phases of the solitons. Detailed asymptotic analysis also shows that when solitons undergo multiple collisions, there exists the exciting possibility of shape restoration of at least one soliton during interactions of more than two solitons represented by three- and higher-order soliton solutions. From an application point of view, we have shown from the asymptotic expressions how the amplitude (intensity) redistribution can be written as a generalized linear fractional transformation for the N-component case. Also we indicate how the multisolitons can be reinterpreted as various logic gates for suitable choices of the soliton parameters, leading to possible multistate logic. In addition, we point out that the various recently studied partially coherent solitons are just special cases of the bright soliton solutions exhibiting shape-changing collisions, thereby explaining their variable profile and shape variation in collision process.

A different kind of shape changing (intensity redistribution) collision with potential application to signal amplification is identified in the integrable -coupled nonlinear Schrödinger (CNLS) equations with mixed signs of focusing- and defocusing-type nonlinearity coefficients. The corresponding soliton solutions for the N=2 case are obtained by using Hirota's bilinearization method. The distinguishing feature of the mixed sign CNLS equations is that the soliton solutions can both be singular and regular. Although the general soliton solution admits singularities we present parametric conditions for which nonsingular soliton propagation can occur. The multisoliton solutions and a generalization of the results to the multicomponent case with arbitrary N are also presented. An appealing feature of soliton collision in the present case is that all the components of a soliton can simultaneously enhance their amplitudes, which can lead to a different kind of amplification process without induced noise.

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.