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Integrable differential equations and coverings of elliptic curves. (English. Russian original) Zbl 0547.35109

Math. USSR, Izv. 22, 357-377 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 2, 384-406 (1983).
The author considers the equation \([\partial /\partial x-M,\partial /\partial t-N]=0\) of coefficients of rational matrix-functions M and N on elliptic curves E, depending on x,\(t\in R\). This includes important examples, such as the Landau-Lifshitz equation with respect to the vector valued function \(u(x,t)=(u_ 1,u_ 2,u_ 3):\quad\partial u/\partial t=u\times\partial^ 2u/\partial x^ 2+u\times Ju\) and also the systems of equations \(\partial u/\partial t=u\times Ju,\quad\partial v/\partial x=v\times Ju,\) J is the diagonal matrix with constants \(j_ k\). The main aim is the construction of some algebraical approach to the theory of these equations, in particular, to develop the well known Bäcklund- Darboux transformation.
Reviewer: A.Dzhuraev

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35L65 Hyperbolic conservation laws
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37C10 Dynamics induced by flows and semiflows
14H52 Elliptic curves
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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