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  • 1980-1984  (5)
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  • 1
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This work deals with Bäcklund transformations for the principal SL(n, ℂ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a “soliton correlation matrix” which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Integrable 1+1 dimensional systems associated to linear first-order matrix equations meromorphic in a complex parameter, as formulated by Zakharov, Mikhailov, and Shabat [1−3] (ZMS) are analyzed by a new method based upon the “soliton correlation matrix” (M-matrix). The multi-Bäcklund transformation, which is equivalent to the introduction of an arbitrary number of poles in the ZMS dressing matrix, is expressed by a pair of matrix Riccati equations for theM-matrix. Through a geometrical interpretation based upon group actions on Grassman manifolds, the solution of this system is explicitly determined in terms of the solutions to the ZMS linear system. Reductions of the system corresponding to invariance under finite groups of automorphisms are also solved by reducing theM-matrix suitably so as to preserve the class of invariant solutions.
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  • 3
    ISSN: 1573-0530
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Abstract It is shown how the reduction method applied to a 2×2 Zakharov-Shabat system with appropriate meromorphic structure leads to the Thirring model as integrability conditions. Reducing the generic soliton-generating multi-Bäcklund transformations, the general multi-soliton solutions are explicitly derived.
    Notes: Resumé It est démontré que la méthode de réduction appliquée à un système d'ordre 2×2 du type Zakharov-Shabat, muni d'une structure méromorphe appropriée, amène au modèle classique de Thirring; ce dernier étant de fait la condition d'intégrabilité du précédent système. Résuisant les transformations de Bäcklund génératrices de solitons, les solutions multi-solitons sont dérivées de manière explicite.
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  • 4
    ISSN: 1573-0530
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The classification of systems of nonlinear ordinary differential equations with superposition principles is reduced to a classification of transitive primitive Lie algebras. Each system can be associated with the transitive primitive action of a Lie group G on a homogenous space G/H, where H is a maximal subgroup of G. The equations can have specific polynomial or rational nonlinearities.
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  • 5
    ISSN: 1573-0530
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Using the invariant geometrical interpretation of gauge and Higgs fields, a simple derivation is given of the dimensional reduction procedure. The underlying assumption with regard to the Riemannian structure, group orbits and invariant connection are clarified and the critical points of the Higgs potential are shown to have a natural geometrical interpretation.
    Type of Medium: Electronic Resource
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