Greek Laughter : a Study of Cultural Psychology from Homer to Early Christianity

Frontmatter......Page 1
1 Introduction......Page 12
2 Various levels of integrability for PDEs, de.nitions......Page 13
3 Importance of the singularities: a brief survey of the theory of Painlev´e......Page 20
4.1 Singular manifold variable . and expansion variable χ......Page 22
4.2 The WTC part of the Painlev´e test for PDEs......Page 25
4.3 The various ways to pass or fail the Painlev´e test for PDEs......Page 28
5 Ingredients of the “singular manifold method”......Page 29
5.2 Transposition of the ODE situation to PDEs......Page 30
5.3 The singular manifold method as a singular part transformation......Page 31
5.5 Choices of Lax pairs and equivalent Riccati pseudopotentials......Page 32
6 The algorithm of the singular manifold method......Page 35
6.1 Where to truncate, and with which variable?......Page 38
7.1 Integrable equations with a second order Lax pair......Page 40
7.2 Integrable equations with a third order Lax pair......Page 46
7.3 Nonintegrable equations, second scattering order......Page 60
7.4 Nonintegrable equations, third scattering order......Page 63
8.1 The constant level term does not de.ne a BT......Page 64
9 The singular manifold method applied to two-family PDEs......Page 65
9.1 Integrable equations with a second order Lax pair......Page 66
9.2 Integrable equations with a third order Lax pair......Page 70
9.3 Nonintegrable equations, second and third scattering order......Page 71
10 Singular manifold method versus reduction methods......Page 80
11 Truncation of the unknown, not of the equation......Page 83
12 Birational transformations of the Painlev´e equations......Page 85
13 Conclusion, open problems......Page 87
1 Introduction: The tensorial approach and the birth of the method of Poisson pairs......Page 95
1.1 The Miura map and the KdV equation......Page 96
1.2 Poisson pairs and the KdV hierarchy......Page 98
1.3 Invariant submanifolds and reduced equations......Page 100
1.4 The modi.ed KdV hierarchy......Page 104
2 The method of Poisson pairs......Page 106
3.1 Lie–Poisson manifolds......Page 111
3.2 Polynomial extensions......Page 112
3.3 Geometric reduction......Page 113
3.4 An explicit example......Page 114
3.5 A more general example......Page 118
4.1 Poisson pairs on a loop algebra......Page 119
4.2 Poisson reduction......Page 120
4.3 The GZ hierarchy......Page 122
4.4 The central system......Page 123
4.5 The linearization process......Page 125
4.6 The relation with the Sato approach......Page 127
5.1 Lax representation......Page 130
5.2 First example......Page 132
5.3 The generic stationary submanifold......Page 134
6 Darboux–Nijenhuis coordinates and separability......Page 135
6.1 The Poisson pair......Page 136
6.2 Passing to a symplectic leaf......Page 138
6.3 Darboux–Nijenhuis coordinates......Page 140
6.4 Separation of variables......Page 141
1 Introduction......Page 147
2 Integrability by the singularity approach......Page 148
4.1 Nonlinear ordinary di.erential equations......Page 149
4.2 Nonlinear partial di.erential equations......Page 152
5 Lax Pair and Darboux transformation......Page 153
5.1 Second order scalar scattering problem......Page 154
5.2 Third order scalar scattering problem......Page 155
5.3 A third order matrix scattering problem......Page 156
6 Di.erent truncations in Painlev´e analysis......Page 157
7 Method for a one-family equation......Page 159
9.1 First example: KdV equation......Page 161
9.2 Second example: MKdV and sine-Gordon equations......Page 163
10.1 Sawada-Kotera, KdV5, Kaup-Kupershmidt equations......Page 166
10.2 Painlev´e test......Page 167
10.4 Truncation with a third order Lax pair......Page 168
10.5 B¨acklund transformation......Page 169
10.6 Nonlinear superposition formula for Sawada-Kotera......Page 170
10.7 Nonlinear superposition formula for Kaup-Kupershmidt......Page 171
10.8 Tzitz´eica equation......Page 175
1 Introduction......Page 181
2.1 The Burgers equation......Page 182
2.2 The Korteweg-de Vries equation......Page 183
2.3 The nonlinear Schr¨odinger equation......Page 184
2.4 The Toda equation......Page 185
2.5 Painlev´e equations......Page 186
2.6 Di.erence vs di.erential......Page 187
3.1 The Kadomtsev-Petviashvili equation......Page 190
3.2 The two-dimensional Toda lattice equation......Page 191
3.3 Two-dimensional Toda molecule equation......Page 194
3.4 The Hirota-Miwa equation......Page 195
4.1 Micro-di.erential operators......Page 197
4.2 Introduction of an in.nite number of time variables......Page 199
4.3 The Sato equation......Page 202
4.4 Generalized Lax equation......Page 204
4.5 Structure of tau functions......Page 205
4.6 Algebraic identities for tau functions......Page 210
4.7 Vertex operators and the KP bilinear identity......Page 214
4.8 Fermion analysis based on an in.nite dimensional Lie algebra......Page 217
5.1 q-discrete equations......Page 220
5.2 Special function solution for soliton equations......Page 222
5.3 Ultra discrete soliton system......Page 225
5.4 Trilinear equations......Page 228
1 Introduction......Page 233
2.1 Formulation of the problem......Page 235
2.2 Prolongation of vector .elds and the symmetry algorithm......Page 238
2.3 First example: Variable coe.cient KdV equation......Page 240
2.4 Symmetry reduction for the KdV......Page 242
2.5 Second example: Modi.ed Kadomtsev-Petviashvili equation......Page 245
3.1 Formulation of the problem......Page 248
3.2 Subalgebras of a simple Lie algebra......Page 249
3.3 Example: Maximal subalgebras of o(4, 2)......Page 250
3.4 Subalgebras of semidirect sums......Page 254
3.5 Example: All subalgebras of sl(3,R) classi.ed under the group SL(3,R)......Page 258
4.1 Formulation of the problem......Page 262
4.2 Symmetry reduction for Boussinesq equation......Page 263
4.3 The direct method......Page 264
4.4 Conditional symmetries......Page 266
4.5 General comments......Page 271
5.1 References on nonlinear superposition formulas......Page 273
5.2 References on continuous symmetries of di.erence equations......Page 274
Backmatter......Page 284