Integration of the bihamiltonian systems by dressing method

Abstract: We consider of nonlinear bi-hamiltonian systems of evolution equations in the form

ut = K[u],

(1)

where u = (u₁,...,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]

Lt = [K¢,L],

(2)

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

L j = lj

jt = K¢[u]j,

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].

1. Mitropolskiy Yu.A., Bogolyubov N.N., Prikarpatskiy A.K., Samoylenko V.H. Integrable dynamical systems. Kiev, Naukova dumka. - 1987. - 296 p. (in russian)
2. Sidorenko Yu.M. Method of integration of Lax equations with nonlocal reductions // Dopovidi NAN Ukrainy. - 1999. N9. - p.33-36.
3. Sidorenko Yu. Transformation operators for integrable hierarchies with additional reductions // Proceedings of Institute of Math. of NAS of Ukraine. - 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.- V. 17.- N. 1. - p. 47-58