Direct Methods in Soliton Theory

Ryogo Hirota - Collection Cambridge Tracts in Mathematics

200 pages, parution le 06/10/2004

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Résumé

The bilinear, or Hirota's direct, method was invented in the early 1970s as an elementary means of constructing soliton solutions that avoided the use of the heavy machinery of the inverse scattering transform and was successfully used to construct the multisoliton solutions of many new equations. In the 1980s the deeper significance of the tools used in this method - Hirota derivatives and the bilinear form - came to be understood as a key ingredient in Sato's theory and the connections with affine Lie algebras. The main part of this book concerns the more modern version of the method in which solutions are expressed in the form of determinants and pfaffians. While maintaining the original philosophy of using relatively simple mathematics, it has, nevertheless, been influenced by the deeper understanding that came out of the work of the Kyoto school. The book will be essential for all those working in soliton theory.

Sommaire

Bilinearization of soliton equations
- Solitary waves and solitons
- Nonlinearity and dispersion
- Solutions of nonlinear differential equations
- Linearization of nonlinear differential equations
- Essentials of the direct method
- The D-operator, a new differential operator
- Bilinearization of nonlinear differential equations
- Solutions of bilinear equations
- Transformation from bilinear to nonlinear form
Determinants and pfaffians
- Introduction
- Pfaffians
- Exterior algebra
- Expressions for general determinants and wronskians
- Laplace expansions of determinants and Plücker relations
- Jacobi identities for determinants
- Special determinants
- Pfaffian identities
- Expansion formulae for the pfaffian (a1, a2, 1, 2, . . . , 2n)
- Addition formulae for pfaffians
- Derivative formulae for pfaffians
Structure of soliton equations
- Introduction
- The KP equation
- The BKP equation: pfaffian solutions
- The coupled KP equation
- The two-dimensional Toda lattice equation
- The two-dimensional Toda molecule equation
- Bäcklund transformations
- What is a Bäcklund transformation?
Bäcklund transformations for KdV-type bilinear equations
- The Bäcklund transformation for the KP equation
- The Bäcklund transformation for the BKP equation
- The solution of the modified BKP equation
- The Bäcklund transformation for the two-dimensional Toda equation
- Solution of the two-dimensional modified Toda equation

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	PAPIER
Éditeur(s)	Cambridge University Press
Auteur(s)	Ryogo Hirota
Collection	Cambridge Tracts in Mathematics
Parution	06/10/2004
Nb. de pages	200
Format	15,5 x 23,5
Couverture	Relié
Poids	495g
Intérieur	Noir et Blanc
EAN13	9780521836609
ISBN13	978-0-521-83660-9