Geometry of solitons. (English) Zbl 0987.37072

Terng, Chuu-Lian; Uhlenbeck, Karen

Notices Am. Math. Soc. 47, No. 1, 17-25 (2000).

This survey paper gives a deep and comprehensive insight to the geometrical theory of solitons. Relations with classical differential geometry of surfaces in \({\mathbb R}^3\), especially pseudospherical surfaces, are in the center of the exposition. Basics of Bäcklund transformations, dressing actions, inverse scattering method are nicely presented. The sine-Gordon equation is used to illustrate basic ideas and definitions.

Reviewer: Vassili Gelfreich (Berlin)

MSC:

37K35	Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51	Soliton equations
37-02	Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
53A05	Surfaces in Euclidean and related spaces
53C43	Differential geometric aspects of harmonic maps
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

solitons; inverse scattering; Bäcklund transformation; Lax pair