Yuriy BERKELA
Ivan Franko National University of Lviv,
1 Universytetska Street, Lviv, 79000, UKRAINE
E-mail: yuri@rakhiv.ukrtel.net
Integration of the bihamiltonian systems by dressing
method
Abstract:
We consider of nonlinear bi-hamiltonian systems of evolution
equations in the form
where u = (u_{1},...,u_{m})(x,t) be a smooth vector-function,
and K[u] be a linear functional.
Moving from, so-called, "recursion" Lax representation for system
(1), known at present time for most integrable systems in
dimension (1+1) [1]
where L be a generating (symmetrical-recursion) operator,
K¢ = K¢[u] be a Freshe derivative of functional K[u]
(1), we propose the method of integration of system (1), which is
basing on idea of dressing transformations of Zakharov-Shabat and
Dorboux-Matveev.
Factorization of generating operator L with two
hamiltonian operators L ³ M: L = ML^{-1} admits to describe whole group of reductions
associating linear integro-differential system
where l be a spectral parameter.
Wide classes of exact solutions of system (1) may be obtain as
nonlinear superposition of linear waves, analogically to method of
integration of nonlinear models with integro-differential
Lax-Zakharov-Shabat representations [2-4].
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with nonlocal reductions
// Dopovidi NAN Ukrainy. - 1999. N9. - p.33-36.
3. Sidorenko Yu. Transformation operators for integrable
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- 2002.- V.43, Part 1. - pp. 352-357
4. Berkela Yu.Yu., Sidorenko Yu.M. The exact solutions of
some multicomponent integrable models // Mat. Studii.- 2002.-
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