V. S. Gerdjikov

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In the past decades there has been significant progress in the theory of the nonlinear evolution equations that could be solved through some inverse scattering techniques (soliton equations or completely integrable equations). These equations have many applications to various physical theories (both classical and quantum) such as hydrodynamics, plasma(More)
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(More)
The squared eigenfunctions of the spectral problem associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the Camassa-Holm hierarchy as a Generalised Fourier transform. The main result of this work is the derivation of the completeness relation for the squared solutions(More)
The asymptotic regimes of the N-site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymp-totically free propagation (the only possible regime for the RTC), CTC allow bound(More)
An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard Inverse Scattering Transform lies in the fact that we have a weighted spectral problem. We(More)
The analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is 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. Some of the reduced systems find(More)
Using the Karpman-Solov'ev method we derive the equations for the two-soliton adiabatic interaction for solitons of the modified nonlinear Schrödinger equation (MNSE). Then we generalize these equations to the case of N interacting solitons with almost equal velocities and widths. On the basis of this result we prove that the N MNSE-soliton train(More)
The reductions of the multi-component nonlinear Schrödinger (MNLS) type models related to C.I and D.III type symmetric spaces are studied. We pay special attention to the MNLS related to the sp(4), so(10) and so(12) Lie algebras. The MNLS related to sp(4) is a three-component MNLS which finds applications to Bose-Einstein condensates. The MNLS related to(More)