The Direct Method in Soliton Theory

Hirota R., Nagai A. (Ed), Nimmo J. (Ed)978-0-511-21663-3

Hirota invented his bilinear or direct method in the early 1970s as a way of constructing soliton solutions without dealing with the cumbersome inverse scattering transform. His invention has since come to the Kyoto School and became connected with affine Lie algebras, but here Hirota explains the more modern version of the method, in which solutions are expressed in the form of determinants and pfaffians. Hirota covers elementary properties of linear and nonlinear waves and eave expressions, the properties of determinants and pfaffians, and gives examples of soliton equations, finishing with a discussion of Bäcklund transforms.

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Table of contents :
Half-title……Page 2
Title……Page 4
Copyright……Page 5
Contents……Page 6
Foreword……Page 8
Preface……Page 10
1.1 Solitary waves and solitons……Page 14
1.2.1 Linear nondispersive waves……Page 15
1.2.2 Linear dispersive waves……Page 16
1.2.3 Nonlinear nondispersive waves……Page 17
1.2.4 Nonlinear dispersive waves……Page 18
1.3 Solutions of nonlinear differential equations……Page 22
1.4.1 The Riccati equation……Page 25
1.4.2 The Burgers equation……Page 26
1.4.3 The Liouville equation……Page 28
1.4.4 Two-wave interaction equations……Page 30
1.5 Essentials of the direct method……Page 32
1.6 The D-operator, a new differential operator……Page 40
1.7.1 Rational transformation……Page 50
1.7.2 Logarithmic transformation……Page 54
1.7.3 Bi-logarithmic transformation……Page 56
1.8 Solutions of bilinear equations……Page 59
1.9.2 Logarithmic transformation……Page 71
1.9.3 Bi-logarithmic transformation……Page 72
2.1 Introduction……Page 73
2.2 Pfaffians……Page 74
2.3 Exterior algebra……Page 77
2.4 Expressions for general determinants and wronskians……Page 79
2.5.1 Laplace expansions of determinants……Page 83
2.5.2 Plücker relations……Page 86
2.6.1 Cofactors……Page 90
2.6.2 Jacobi identities for determinants [33]……Page 92
2.7.1 Perfect square formula (i)……Page 97
2.7.2 Perfect square formula (ii)……Page 98
2.7.3 Bordered determinants……Page 100
2.8 Pfaffian identities……Page 105
2.9 Expansion formulae for the pfaffian (a1, a2, 1, 2,…, 2n)……Page 110
2.10 Addition formulae for pfaffians……Page 112
2.11 Derivative formulae for pfaffians……Page 114
3.1 Introduction……Page 123
3.2.1 Wronskian solutions……Page 124
3.2.2 Grammian solutions……Page 134
3.3 The BKP equation: pfaffian solutions……Page 140
3.4.1 Wronski-type pfaffian solutions……Page 146
3.4.2 Gramm-type pfaffian solutions……Page 152
3.5.1 Wronskian solutions……Page 155
3.5.2 Grammian solutions……Page 158
3.6.1 Bi-directional wronskian solutions……Page 162
3.6.2 Double wronskian solutions……Page 167
4.1 What is a Bäcklund transformation?……Page 170
4.2 Bäcklund transformations for KdV-type bilinear equations……Page 174
4.2.1 Inverse scattering formulation……Page 177
4.2.2 The modified KdV (mKdV) equation……Page 178
4.2.3 The Miura transformation……Page 179
4.3 The Bäcklund transformation for the KP equation……Page 182
4.3.1 Wronskian expression……Page 183
4.3.2 Grammian expression……Page 186
4.4 The Bäcklund transformation for the BKP equation……Page 188
4.4.1 Inverse scattering form……Page 189
4.5 The solution of the modified BKP equation……Page 190
4.6 The Bäcklund transformation for the two-dimensional Toda equation……Page 192
4.6.1 Lax pair……Page 193
4.6.2 The modified Toda equation……Page 194
4.6.3 Miura transformation……Page 196
4.7.1 Structure of the Bäcklund transformation for the Toda lattice equation……Page 197
Solution with the same number of solitons but a different phase……Page 198
Solution with one more soliton……Page 200
4.7.2 Structure of the Bäcklund transformation for the Toda molecule equation……Page 201
Afterword……Page 205
References……Page 208
Index……Page 211