The Direct Method in Soliton TheoryCambridge University Press, 22 июл. 2004 г. - Всего страниц: 200 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. |
Содержание
Bilinearization of soliton equations | 1 |
9 | 58 |
5 | 70 |
6 | 77 |
8 | 92 |
Structure of soliton equations | 110 |
101 | 141 |
Bäcklund transformations | 157 |
Afterword | 192 |
Часто встречающиеся слова и выражения
algebra aN,N an.n b₁ b₂ Bäcklund transformation Bäcklund transformation formulae bilinear form bilinear KP equation constant coupled KP equation D-operator dependent variable transformation derivatives direct method Dx(D equa exact solution exp n2 expansion formula grammian Hirota Jacobi identity KdV equation KP equation Laplace expansion lattice equation Lax pair left-hand side linear differential equations Liouville equation matrix Maya diagram expression Mikio Sato molecule equation N-soliton solution N+1 N+2 N+3 nonlinear differential equations nonlinear partial differential notation obtain P₁ parameters partial differential equation perturbation method pfaffian entries pfaffian identity Phys Plücker relation Remark rewritten right-hand side satisfy the linear solitary wave soliton equations tion Tn+1 Toda lattice two-dimensional two-soliton solution Txxx u₁ uxxx wronskian zero дх дхп Эх მ მ