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- V. S. Gerdjikov, Gaetano Vilasi, Alexandar B. Yanovski, Alexandar B. Yanovski
- 2013

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, plasmaâ€¦ (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-waveâ€¦ (More)

Abstract. The class of nonlinear evolution equations (NLEE) â€“ gauge equivalent to the N-wave equations related to the simple Lie algebra g are derived and analyzed. They are written in terms of S(x, t) âˆˆ g satisfying r = rank g nonlinear constraints. The corresponding Lax pairs and the time evolution of the scattering data are found. The Zakharovâ€“Shabatâ€¦ (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 solutionsâ€¦ (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 boundâ€¦ (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. Weâ€¦ (More)

This is a review of two of the fundamental tools for analysis of soliton equations: i) the algebraic ones based on Kac-Moody algebras, their central extensions and their dual algebras which underlie the Hamiltonian structures of the NLEE; ii) the construction of the fundamental analytic solutions (FAS) of the Lax operator and the Riemann-Hilbert problemâ€¦ (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 Nâ€“wave interactions related to lowâ€“rank simple Lie algebras g can be embedded also in the Weyl group of g. Some of the reduced systems findâ€¦ (More)

- V. S. Gerdjikov, E. V. Doktorov, Junzhong Yang
- Physical review. E, Statistical, nonlinear, andâ€¦
- 2001

Using the Karpman-Solov'ev method we derive the equations for the two-soliton adiabatic interaction for solitons of the modified nonlinear SchrÃ¶dinger 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 trainâ€¦ (More)

The reductions of the multi-component nonlinear SchrÃ¶dinger (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 toâ€¦ (More)