Browsing by Author "Mikhailov A.V."

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    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.
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    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.
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    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.
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    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.