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Soliton surfaces and their applications (soliton geometry from spectral problems). (English) Zbl 0583.53017
Geometric aspects of the Einstein equations and integrable systems, Proc. 6th Conf., Scheveningen/Neth. 1984, Lect. Notes Phys. 239, 154-231 (1985).

[For the entire collection see Zbl 0572.00009.]

Soliton theory here is to be understood as a theory of special classes of completely integrable nonlinear systems of partial differential equations. The main purpose of the paper is to clarify the connections between soliton structures and the differential geometry of certain manifolds, the so called soliton surfaces, which are generalizations of the pseudo-spherical surfaces related to the sine-Gordon equation. The crucial points are: soliton systems can be represented as integrability conditions of some linear systems of partial differential equations, such integrability conditions can be given a differential geometric meaning, as being closely related to the Gauss-Mainardi-Codazzi-Ricci equations of submanifolds of constant sectional curvature of an affine space with non- degenerate but not necessarily positive definite metric bilinear form. The paper works with soliton systems of the Z.S.-A.K.N.S. (Zakharov- Shabat-Ablowitz-Kaup-Newell-Segur) class, analyses the relation to geometry and contains detailed descriptions of several examples.

Reviewer: K.Horneffer

MSC:
53B25Local submanifolds
53B50Applications of local differential geometry to physics
35G99General higher order PDE