Abstract.The reductions of the integrable N -wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analyzed. The Zakharov-Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one soliton solutions of the corresponding N -wave equations and their reductions are studied. We show that to each soliton solution one can relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave equations related to so(5) which find applications in Stockes-anti-Stockes wave generation.
Abstract. We consider travelling periodic and quasi-periodic wave solutions of a set of coupled nonlinear Schrödinger equations. In fibre optics these equations can be used to model single mode fibers with strong birefingence and two-mode optical fibers. Recently these equations appear as model, which describe pulse-pulse interaction in wavelength-division-multiplexed channels of optical fiber transmission systems. Two phase quasi-periodic solutions for integrable Manakov system are given in terms of two dimensional Kleinian functions. The reduction of quasi-periodic solutions to elliptic functions is discussed. New solutions in terms generalized Hermite polynomials, which are associated with two-gap TreibichVerider potentials are found.
Abstract. The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem. We show how the second order reductions of the N -wave interactions related to low-rank simple Lie algebras g can be embedded also in the Weyl group of g. This allows us to display along with the well known ones a number of new types of integrable N -wave systems. Some of the reduced systems find applications to nonlinear optics.
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