Browsing by Author "Valchev T.I."
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Item A Generic Nonlinear Evolution Equation of Magnetic Type I. Reductions(2023-01-01) Valchev T.I.The present report is dedicated to a completely integrable nonlinear evolution equation that generalizes Heisenberg’s ferromagnet equation. That generalization has a linear bundle Lax pair in pole gauge related to a homogeneous space. A few local and nonlocal reductions of the generic matrix equation are considered.Item Bose-Einstein condensates with F=1 and F=2. Reductions and soliton interactions of multi-component NLS models(2009-12-01) Gerdjikov V.S.; Kostov N.A.; Valchev T.I.We analyze a class of multicomponent nonlinear Schrödinger equations (MNLS) related to the symmetric BD.I-type symmetric spaces and their reductions. We briefly outline the direct and the inverse scattering method for the relevant Lax operators and the soliton solutions. We use the Zakharov-Shabat dressing method to obtain the two-soliton solution and analyze the soliton interactions of the MNLS equations and some of their reductions. © 2009 Copyright SPIE - The International Society for Optical Engineering.Item Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions(2016-02-01) Valchev T.I.We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m + n)/S(U(m) × U(n)). The simplest representative of the corresponding integrable hierarchy is given by a multi-component Kaup-Newell derivative nonlinear Schrödinger equation which serves as a motivational example for our general considerations. We extensively discuss how one can apply Zakharov-Shabat's dressing procedure to derive reflectionless potentials obeying zero boundary conditions. Those could be used for one to construct fast decaying solutions to any nonlinear equation belonging to the same hierarchy. One can distinguish between generic soliton type solutions and rational solutions.Item Exact solutions for equations of Bose-Fermi mixtures in one-dimensional optical lattice(2007-01-01) Kostov N.A.; Gerdjikov V.S.; Valchev T.I.We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose- Fermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.Item Hermitian and pseudo-hermitian reduction of the GMV auxiliary system. Spectral properties of the recursion operators(2019-01-01) Yanovski A.B.; Valchev T.I.We consider simultaneously two different reductions of a Zakharov-Shabat’s spectral problem in pole gauge. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of the Recursion Operators related to the afore-mentioned spectral problem with arbitrary constant asymptotic values of the potential functions. In doing this, we take into account the discrete spectrum of the scattering operator. Having in mind the applications to the theory of the soliton equations associated to the GMV systems, we show how these expansions modify depending on the symmetries of the functions we expand.Item N-wave equations with orthogonal algebras: Z2and Z2× Z2reductions and soliton solutions(2007-01-01) Gerdjikov V.S.; Kostov N.A.; Valchev T.I.We consider N-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first Z2-reduction is the canonical one. We impose a second Z2-reduction and consider also the combined action of both reductions. For all three types of N-wave equations we construct the soliton solutions by appropriately modifying the Zakharov-Shabat dressing method. We also briefly discuss the different types of one-soliton solutions. Especially rich are the types of one-soliton solutions in the case when both reductions are applied. This is due to the fact that we have two different configurations of eigenvalues for the Lax operator L: doublets, which consist of pairs of purely imaginary eigenvalues, and quadruplets. Such situation is analogous to the one encountered in the sine-Gordon case, which allows two types of solitons: kinks and breathers. A new physical system, describing Stokes-anti Stokes Raman scattering is obtained. It is represented by a 4-wave equation related to the B2algebra with a canonical Z2reduction.Item On the quadratic bundles related to Hermitian symmetric spaces(2013-01-01) Valchev T.I.Here we develop the direct scattering problem for quadratic bundles associated to Hermitian symmetric spaces. We adapt the dressing method for quadratic bundles which allows us to find special solutions to multicomponent derivative Schrödinger equation for instance. The latter is an infinite dimensional Hamiltonian system possessing infinite number of integrals of motion. We demonstrate how one can derive them by block diagonalization of the corresponding Lax pair.Item Polynomial bundles and Generalised Fourier transforms for integrable equations on A.III-type symmetric spaces(2011-01-01) Gerdjikov V.S.; Grahovski G.G.; Mikhailov A.V.; Valchev T.I.A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.Item Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties(2018-04-03) Yanovski A.B.; Valchev T.I.We consider an auxiliary spectral problem originally introduced by Gerdjikov, Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitian reduction which is extensively studied here for the first time. We describe the integrable hierarchies of both systems in a parallel way and construct recursion operators. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of recursion operators. This permits us to obtain the expansions for both GMV systems with arbitrary constant asymptotic values of the potential functions in the auxiliary linear problems.Item Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions(2018-07-03) Valchev T.I.; Yanovski A.B.This paper is a continuation of our previous work in which we studied a sl (3, ℂ) Zakharov-Shabat type auxiliary linear problem with reductions of Mikhailov type and the corresponding integrable hierarchy of nonlinear evolution equations. Now, we shall demonstrate how one can construct special solutions over constant back- ground through Zakharov-Shabat’s dressing technique. That approach will be illustrated on the example of the generalized Heisenberg ferromagnet equation related to the linear problem for sl (3, ℂ). In doing this, we shall discuss the differences between the Hermitian and pseudo-Hermitian cases.Item Rational bundles and recursion operators for integrable equations on A.III-type symmetric spaces(2011-06-01) Gerdjikov V.S.; Grahovski G.G.; Mikhailov A.V.; Valchev T.I.We analyze and compare methods for constructing the recursion operators for a special class of integrable nonlinear differential equations related to symmetric spaces of the type A. III in Cartan's classification and having additional reductions. © 2011 Pleiades Publishing, Ltd.Item Recursion operators and reductions of integrable equations on symmetric spaces(2010-01-01) Gerdjikov V.S.; Mikhailov A.V.; Valchev T.I.We study certain classes of integrable nonlinear differential equations related to the type symmetric spaces. Our main examples concern equations related to A.III-type symmetric spaces. We use the Cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic Lax operators to this symmetric space. Next we outline the spectral theory of the reduced Lax operator L and construct its fundamental analytic solutions. Analyzing the Wronskian relations we introduce the 'squared solutions' of L and derive the recursion operators by three different methods.Item Reductions of integrable equations on A. III-type symmetric spaces(2010-10-29) Gerdjikov V.S.; Mikhailov A.V.; Valchev T.I.We study a class of integrable nonlinear differential equations related to the A. III-type symmetric spaces. These spaces are realized as factor groups of the form SU (N)/S(U (N-k) × U (k)). We use the Cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic Lax operators to this symmetric space. The symmetries of the Lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding Riemann-Hilbert data. © 2010 IOP Publishing Ltd.Item Solutions of multi-component NLS models and Spinor Bose-Einstein condensates(2009-07-15) Gerdjikov V.S.; Kostov N.A.; Valchev T.I.Three- and five-component nonlinear Schrödinger-type models, which describe spinor Bose-Einstein condensates (BEC's) with hyperfine structures F = 1 and F = 2, respectively, are studied. These models for particular values of the coupling constants are integrable by the inverse scattering method. They are related to symmetric spaces of BD.I-type ≃ SO(2r + 1) / SO(2) × SO(2r -1) for r = 2 and r = 3. Using conveniently modified Zakharov-Shabat dressing procedure we obtain different types of soliton solutions. © 2008 Elsevier B.V. All rights reserved.