Browsing by Author "Valchev T."

Now showing 1 - 7 of 7
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    Description of all conformally invariant differential operators, acting on scalar functions
    (2007-12-01) Nikolov P.; Valchev T.
    We present an algorithm to obtain all conformally invariant differential operators acting on scalar functions and taking values on scalar functions as well. There are two cases under consideration: Operators acting on functions defined on the Euclidean space R4 and on the Minkowski space. In both cases each conformally invariant operator of order k is nonlinear and is a function of a finite set of functionally independent invariant operators of order up to k. In particular, the second order independent operators are only three and we give their explicit realization. The applied technique is based on the jet bundle formalism, analysis of the group action, and a dimensional reduction. As a simple illustration of this method we consider the case of differential operators between analytic functions on C which are invariant under the action of the modular group. A power series which generates all functionally independent invariant operators is derived. © 2007 American Institute of Physics.
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    Multicomponent nonlinear evolution equations of the Heisenberg ferromagnet type: Local versus nonlocal reductions
    (2021-01-01) Valchev T.
    This work is dedicated to systems of matrix nonlinear evolution equations related to Hermitian symmetric spaces of the type A.III. The systems under consideration generalize the 1 + 1 dimensional Heisenberg ferromagnet equation in the sense that their Lax pairs are linear bundles in pole gauge like for the original Heisenberg model. Here we present certain local and nonlocal reductions. A local integrable deformation and some of its reductions are discussed as well.
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    On generalized Fourier transform for Kaup-Kupershmidt type equations
    (2010-01-01) Valchev T.
    We develop the Fourier transform interpretation of the inverse scattering method for nonlinear integrable evolution equations associated with a ℤ3 reduced Zakharov-Shabat system for the Lie algebra sl(3,C). A simple representative of this integrable hierarchy is the well-known Kaup-Kupershmidt equation. Our results admit a natural extention for nonlinear equations connected to a deeply reduced Zakharov-Shabat system related to an arbitrary simple Lie algebra.
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    On Mikhailov's reduction group
    (2015-05-26) Valchev T.
    We propose a generalization of the notion of reduction group which provides group-theoretical tools to study in a uniform way certain classes of nonlocal S-integrable equations like Ablowitz-Musslimani's nonlocal Schrödinger equation. Another benefit of the generalization to be presented here is that it supplies us with a systematic approach to construct solutions to S-integrable equations with prescribed symmetries.
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    On nonlocal reductions of a generalized Heisenberg ferromagnet equation
    (2019-10-02) Valchev T.; Myrzakulov R.; Nugmanova G.; Yesmakhanova K.
    We study nonlocal reductions of a coupled system of equations in 1 + 1 dimensions of the Heisenberg ferromagnet type. The system under consideration is completely integrable through inverse scattering transform and has a Lax pair related to a linear bundle in pole gauge. We describe the integrable hierarchy of nonlinear equations related to our system in terms of generating operators. We present some special solutions associated with four distinct discrete eigenvalues of scattering operator. Using the Lax pair diagonalization method, we derive recurrence formulas for the conserved densities and find the first two simplest densities.
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    On soliton interactions for the hierarchy of a generalised Heisenberg ferromagnetic model on SU(3)/S(U(1)×U(2)) symmetric space
    (2012-03-01) Gerdjikov V.; Grahovski G.; Mikhailov A.; Valchev T.
    We consider an integrable hierarchy of nonlinear evolution equations (NLEE) related to linear bundle Lax operator L. The Lax representation is ℤ2× ℤ2reduced and can be naturally associated with the symmetric space SU(3)/S(U(1) × U(2)). The simplest nontrivial equation in the hierarchy is a generalization of Heisenberg ferromagnetic model. We construct the N-soliton solutions for an arbitrary member of the hierarchy by using the Zakharov-Shabat dressing method with an appropriately chosen dressing factor. Two types of soliton solutions: quadruplet and doublet solitons are found. The one-soliton solutions of NLEEs with even and odd dispersion laws have different properties. In particular, the one-soliton solutions for NLEEs with even dispersion laws are not traveling waves while their velocities and amplitudes are time dependent. Calculating the asymptotics of the N-soliton solutions for t → ± ∞ we analyze the interactions of quadruplet solitons.
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    Remarks on quadratic bundles related to Hermitian symmetric spaces
    (2014-01-01) Valchev T.
    We consider quadratic bundles related to Hermitian symmetric spaces of the type SU(m+n)/S(U(m)×U(n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.