This book presents the foundations of the inverse scattering method and its applications to the theory of solitons in such a form as we understand it in Leningrad. The concept of solitonwas introduced by Kruskal and Zabusky in 1965. A soliton (a solitary wave) is a localized particle-like solution of a nonlinear equation which describes excitations of finite energy and exhibits several characteristic features: propagation does not destroy the profile of a solitary wave; the interaction of several solitary waves amounts to their elastic scat­ tering, so that their total number and shape are preserved. Occasionally, the concept of the soliton is treated in a more general sense as a localized solu­ tion of finite energy. At present this concept is widely spread due to its universality and the abundance of applications in the analysis of various processes in nonlinear media. The inverse scattering method which is the mathematical basis of soliton theory has developed into a powerful tool of mathematical physics for studying nonlinear partial differential equations, almost as vigoraus as the Fourier transform. The book is based on the Hamiltonian interpretation of the method, hence the title. Methods of differential geometry and Hamiltonian formal­ ism in particular are very popular in modern mathematical physics. It is precisely the general Hamiltonian formalism that presents the inverse scat­ tering method in its most elegant form. Moreover, the Hamiltonian formal­ ism provides a link between classical and quantum mechanics.

Ludwig D. Faddeev was born in Leningrad, USSR in 1934. He graduated from the Leningrad State University in 1956 and received his Ph.D. from there in 1959. Since 1959 he has been affiliated with the Leningrad branch of Steklov Mathematical Institute and was its Director from 1976 to 2000. Currently Faddeev is Director of the Euler International Mathematical Institute in St. Petersburg, Russia, and Academician-Secretary of the Mathematics Division of the Russian Academy of Sciences. He was President of the International Mathematical Union during1986-1990. 

Faddeevs principal interests and contributions cover the large area of mathematical physics. They include, in chronological order, quantum scattering theory, spectral theory of automorphic functions, quantization of Yang-Mills theories, Hamiltonian methods in classical and quantum integrable systems, quantum groups and quantum integrable systems on a lattice. Faddeevs work laid a mathematical foundation for functional methods in quantum gauge theories. A great deal of his work was directed towards development of Hamiltonian methods in classical and quantum field theories.

  

 

Leon A. Takhtajan was born in Erevan, Republic of Armenia of the USSR, in 1950. He was schooled in Leningrad, graduated from the Leningrad State University in 1973, and received his Ph.D. from the Leningrad branch of Steklov Mathematical Institute in 1975, with which he was affiliated during1973-1998. Since 1992 he has been Professor of Mathematics at Stony Brook University, USA.

Takhtajans principal interests and contributions are in the area of mathematical physics. They include classical and quantum integrable systems, quantum groups, Weil-Petersson geometry of moduli spaces of Riemann surfaces and moduli spaces of vector bundles, and application of quantum methods to algebraic and complex analysis. His work, together withL.D. Faddeev and E.K. Sklyanin, led to the development of the quantum inverse scattering method from which the theory of quantum groups was born.

The Nonlinear Schrödinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples and Their General Properties.- Fundamental Continuous Models.- Fundamental Models on the Lattice.- Lie-Algebraic Approach to the Classification and Analysis of Integrable Models.- Conclusion.- Conclusion. 

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