Nonlinear Schrödinger equation and N-soliton interactions: Generalized Karpman-Solov'ev approach and the complex Toda chain

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Abstract
A method for the description of the N-soliton interaction, which generalizes in a natural way the Karpman-Solov'ev one for the nonlinear Schrödinger (NLS) equation, is proposed. Using it, we derive a nonlinear system of equations describing the dynamics of the parameters of N well separated solitons with nearly equal amplitudes and velocities. Next we study an exhaustive list of perturbations, relevant for nonlinear optics, which include linear and nonlinear dispersive and dissipative terms, effects of sliding filters, amplitude and phase modulation, etc. We prove that the linear perturbations affect each of the solitons separately, while the nonlinear ones also lead to additional interactive terms between neighboring solitons. Under certain approximations we show that the N-soliton interaction for the unperturbed NLS equation is described by the complex Toda chain (CTC) with N nodes, which is a completely integrable dynamical system with 2N degrees of freedom. A comparison made by numeric simulation shows that CTC gives an adequate description for the soliton interactions for a number of choices of the initial conditions.