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35Qxx Partial differential equations -- Equations of mathematical physics and other areas of application

35P25 Partial differential equations -- Spectral theory and eigenvalue problems -- Scattering theory

35R30 Partial differential equations -- Miscellaneous topics -- Inverse problems

15A60 Linear and multilinear algebra; matrix theory -- Basic linear algebra -- Norms of matrices, numerical range, applications of functional analysis to matrix theory En-ligne: Springerlink | Zentralblatt | MathSciNet

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This book is devoted to the inverse scattering technique applied to the analysis of the asymptotic behaviour for large time of solutions of soliton problems for various classes of nonlinear partial differential equations. More exactly the main purpose of this book is to give a complete and rigorous description of the emergence of solitons from various classes of equations on coordinate regions where the dispersive component of the solution is sufficiently small.

The book contains 8 chapters. Chapter 1 presents a rigorous demonstration of the emergence of solitons from the Korteweg-de Vries initial value problem with arbitrary real initial functions; in chapter 2 existence and uniqueness results for Korteweg-de Vries equation are discussed and asymptotic analysis from chapter 1 is extended onto more complicated and interesting cases. Chapter 3 deals with multisoliton solutions to Korteweg-de Vries equations. Chapter 4 deals with Zakharov-Shabat systems and the question how well a real potential in such a system is approximated by its reflectionless part; the corresponding results are applied to the Korteweg-de Vries equation in chapter 5 and 6; in particular the remarkably similar and important general phase shift formula is presented. Chapter 7 deals with the asymptotic analysis of the sine-Gordon equation. The last chapter 8 is devoted to Zakharov-Shabat systems with complex potential. In a small appendix the author explains that his method cannot apply in the case when the associated group velocity is not of constant sign. (Zentralblatt)

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