Geometric transformations and soliton equations

Details Geometric transformations and soliton equations Geometric transformations and soliton equations

2010-10-27

arxiv.org/abs/1010.5593v1

Advanced Lectures in Mathematics 13 (2010), Handbook of Geometric Analysis (vol II), pp. 301-358

Mathematics

Differential Geometry

Scientific paper

We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting positive elements, (ii) a method---based on (i)---for constructing soliton hierarchies from a symmetric space, (iii) the dressing action of the negative loop subgroup on the space of solutions of the related soliton equation, (iv) classical B\"acklund, Christoffel, Lie, and Ribaucour transformations for surfaces in three-space and their relation to dressing actions, (v) methods for constructing a Lax pair for the Gauss-Codazzi Equation of certain submanifolds that admit Lie transforms, (vi) how soliton theory can be used to generalize classical soliton surfaces to submanifolds of higher dimension and co-dimension.

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