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Nonlinear equations and elliptic curves. (English. Russian original) Zbl 0595.35087
J. Sov. Math. 28, 51-90 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 23, 79-136 (1983).
This paper deals with the algebro-geometric or finite zone solution of noninear equations which can be written as zero curvature equation \(U_ t-V_ x+[U,V]=0\), where U(x,t,\(\lambda)\) and V(x,t,\(\lambda)\) are two matrices and the parameter \(\lambda\) is defined on elliptic curves. The author presents the main idea of global finite-zone integration [see the author, Usp. Mat. Nauk 32, No.6(198), 183-208 (1977; Zbl 0372.35002)] and gives a detailed analysis of applications of this technique to some problem based on the theory of elliptic functions.
The papers divided into three chapters. The first chapter gives a good exposition of the relevant notions and results. The second chapter is concerned with the algebro-geometric spectral theory of the Schrödinger difference operator \[ L\psi_ n=c_ n\psi_{n+1}+V_ n\psi_ n+c_{n-1}\psi_{n-1} \] with periodic coefficients \(c_ n=c_{n+N}\), \(v_ n=v_{n+N}\). The third chapter discusses the Peierls model describing the self-consistent behavior of atoms of a lattice. The author gives a necessary and sufficient condition for a functional related to the model to be extremal on some set and the stability of the extremal. This is a well-written paper, it contains good analysis to a number of typical integrable systems such as the KdV, sine-Gordon, Landau-Lifshits equations, the principal chiral model etc.
Reviewer: G.-Z.Tu

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
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.)
33E05 Elliptic functions and integrals
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