Page 14 - Determinants and Their Applications in Mathematical Physics

Contents xiii

5.8.4 Hankel Determinants with Symmetric Toeplitz
Elements ........................ 231
5.8.5 Hessenberg Determinants with Prime Elements . . 232
5.8.6 Bordered Yamazaki–Hori Determinants — 2 .... 232
5.8.7 Determinantal Identities Related to Matrix
Identities ........................ 233

6 Applications of Determinants in Mathematical Physics 235
6.1 Introduction .......................... 235
6.2 Brief Historical Notes ..................... 236
6.2.1 The Dale Equation .................. 236
6.2.2 The Kay–Moses Equation .............. 237
6.2.3 The Toda Equations .................. 237
6.2.4 The Matsukidaira–Satsuma Equations ....... 239
6.2.5 The Korteweg–de Vries Equation .......... 239
6.2.6 The Kadomtsev–Petviashvili Equation ....... 240
6.2.7 The Benjamin–Ono Equation ............ 241
6.2.8 The Einstein and Ernst Equations ......... 241
6.2.9 The Relativistic Toda Equation ........... 245
6.3 The Dale Equation ....................... 246
6.4 The Kay–Moses Equation ................... 249
6.5 The Toda Equations ...................... 252
6.5.1 The First-Order Toda Equation ........... 252
6.5.2 The Second-Order Toda Equations ......... 254
6.5.3 The Milne-Thomson Equation ............ 256
6.6 The Matsukidaira–Satsuma Equations ........... 258
6.6.1 A System With One Continuous and One
Discrete Variable .................... 258
6.6.2 A System With Two Continuous and Two
Discrete Variables ................... 261
6.7 The Korteweg–de Vries Equation .............. 263
6.7.1 Introduction ...................... 263
6.7.2 The First Form of Solution .............. 264
6.7.3 The First Form of Solution, Second Proof ..... 268
6.7.4 The Wronskian Solution ............... 271
6.7.5 Direct Verification of the Wronskian Solution . . . 273
6.8 The Kadomtsev–Petviashvili Equation ........... 277
6.8.1 The Non-Wronskian Solution ............ 277
6.8.2 The Wronskian Solution ............... 280
6.9 The Benjamin–Ono Equation ................. 281
6.9.1 Introduction ...................... 281
6.9.2 Three Determinants .................. 282
6.9.3 Proof of the Main Theorem ............. 285
6.10 The Einstein and Ernst Equations .............. 287
6.10.1 Introduction ...................... 287