Geometry of solitons. (English) Zbl 0987.37072

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.


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