Classical and Quantum Nonlinear Integrable Systems: Theory and Application

Classical and Quantum Nonlinear Integrable Systems Theory and Applications......Page 1
Contents......Page 3
Preface......Page 8
1.1 Introduction......Page 15
1.2 Nonlinear dispersive waves: Scott Russell phenomenon and solitary waves......Page 16
1.2.1 KdV equation and cnoidal waves and the solitary waves......Page 18
1.3 The Fermi–Pasta–Ulam (FPU) numerical experiments on anharmonic lattices......Page 19
1.3.1 The FPU lattice and recurrence phenomenon......Page 20
1.4.1 Asymptotic analysis and the KdV equation......Page 22
1.5.1 Periodic boundary conditions......Page 24
1.5.2 Initial condition with just two solitary waves......Page 25
1.6 Hirota’s bilinearization method: explicit soliton solutions......Page 27
1.6.2 Two-soliton solution......Page 28
1.6.4 Asymptotic analysis......Page 29
1.7.1 The Miura transformation......Page 30
1.7.2 Galilean invariance and the Schr¨odinger eigenvalue problem......Page 31
1.7.4 Lax pair......Page 32
1.8.1.1 Direct scattering analysis and scattering data at t = 0......Page 33
1.8.1.2 Time evolution of scattering data......Page 34
1.8.1.3 Inverse scattering analysis......Page 36
1.9.1 One-soliton solution (N = 1)......Page 37
1.9.2 Two-soliton solution......Page 38
1.9.3 N-soliton solution......Page 39
1.9.5 Non-reflectionless potentials......Page 41
1.10.1 KdV as a Hamiltonian dynamical system......Page 42
1.10.2 Complete integrability of the KdV equation......Page 43
1.11 Infinite number of conserved densities......Page 44
1.12 Backlund transformations......Page 46
1.14 Lie and Lie–Backlund symmetries......Page 47
1.15 Conclusion......Page 48
References......Page 49
2.1 The classical programme of the Painlev´e school and its achievements......Page 51
2.2 Integrability and Painleve property for partial differential equations......Page 54
2.3 The Painleve test for ODEs and PDEs......Page 56
2.3.1 The Fuchsian perturbative method......Page 61
2.3.2 The non-Fuchsian perturbative method......Page 62
2.4.1 Linearizable equations......Page 63
2.4.2 Auto-Backlund transformation of a PDE: the singular manifold method......Page 64
2.4.3 Single-valued solutions of the Bianchi IX cosmological model......Page 66
2.4.4 Polynomial first integrals of a dynamical system......Page 67
2.4.5 Solitary waves from truncations......Page 68
2.4.6 First-degree birational transformations of Painlev´e equations......Page 69
2.5 Liouville integrability and Painleve integrability......Page 70
2.6 Discretization and discrete Painlev´e equations......Page 71
2.7 Conclusion......Page 72
References......Page 73
3.1 Introduction: who is afraid of discrete systems?......Page 76
3.2 The detector gallery......Page 77
3.2.1 Singularity confinement......Page 78
3.2.2 The perturbative Painleve approach to discrete integrability......Page 80
3.2.3 Algebraic entropy......Page 82
3.2.4 The Nevanlinna theory approach......Page 84
3.3.1 The discrete KdV and its de-autonomization......Page 87
3.3.2 The discrete Painleve equations......Page 89
3.3.3 Linearizable systems......Page 95
3.4.1 Differential-difference systems......Page 98
3.4.2 Ultra-discrete systems......Page 101
3.5 Parting words......Page 103
References......Page 104
4.1.1 The dbar method......Page 107
4.1.2 Coherent structures......Page 108
4.1.2.2 Dromions......Page 109
4.1.3 Organization of this chapter......Page 110
4.2 The KPI equation......Page 111
4.3 The DSII equation......Page 114
References......Page 117
5.1 Introduction......Page 119
5.2.1.1 Ice rule......Page 121
5.2.2 The partition function and the transfer matrix......Page 123
5.2.3.1 The Yang–Baxter relations for six-vertex model......Page 124
5.2.3.2 The coordinate Bethe ansatz......Page 125
5.2.3.3 An example of the eigenvector......Page 126
5.2.4.1 Three phases of the six-vertex model......Page 127
5.2.4.2 Parametrization of the Boltzmann weights......Page 128
5.2.5 Critical singularity in the antiferroelectric regime near the phase boundary......Page 130
5.2.7 Low-lying excited spectrumof the transfermatrix and conformal field theory......Page 132
5.2.7.1 Finite-size corrections......Page 133
5.2.7.2 The free boson: CFT with c = 1......Page 134
5.2.7.3 The XXZ spin chain and CFT with c = 1......Page 136
5.3.1.3 Ashkin–Teller model......Page 137
5.3.2.2 Integrable chiral Potts model......Page 138
5.3.3 The eight-vertex model......Page 139
5.3.4.1 Unrestricted 8V SOS model......Page 140
5.3.4.2 RSOS models......Page 141
5.4.1.1 Derivation of a solution for the six-vertex model......Page 142
5.4.2.1 R-matrix and the L-operator......Page 144
5.4.2.2 Monodromy matrix and the construction of the eigenvector......Page 146
5.5.1.1 The Yang–Baxter equation in operator formalism......Page 148
5.5.1.2 The braid group......Page 149
5.5.2 Quantum groups (Hopf algebras)......Page 150
Appendix. Commuting transfer matrices and the Yang–Baxter equations......Page 151
References......Page 154
6.1 Introduction......Page 159
6.2 Integrable structures in ultralocal models......Page 161
6.2.1 List of well-known ultralocal models......Page 162
6.2.1.1 Models associated with the trigonometric R-matrix (q = ei•, • = ei•)......Page 163
6.2.1.1a Models associated with the twisted trigonometric R-matrix......Page 164
6.2.1.2 Models associated with the rational R-matrix......Page 165
6.3 Unifying algebraic approach in ultralocal models......Page 166
6.3.1 Generation of models......Page 167
6.3.1.1 Models belonging to the trigonometric class......Page 168
6.3.1.1a Models in the twisted trigonometric class......Page 170
6.3.1.2 Models belonging to the rational class......Page 171
6.3.2 Fundamental and regular models......Page 173
6.3.4 Construction of classical models......Page 176
6.4 Integrable statistical systems: vertex models......Page 177
6.5.1 Inhomogeneous models......Page 178
6.5.3 Non-fundamental statistical models......Page 179
6.6 Unified Bethe ansatz solution......Page 180
6.7.1 Braided extensions of QYBE......Page 183
6.7.2.1 Systems with spectral-parameter-less R-matrix......Page 185
6.7.2.3 Models with trigonometric R(•)-matrix......Page 186
6.7.4 Open directions in non-ultralocal models......Page 187
6.8 Concluding remarks......Page 188
References......Page 189
7.1 Introduction......Page 192
7.2 Spin models in one dimension......Page 194
7.3 Ladder models......Page 205
7.4 Concluding remarks......Page 213
References......Page 214
8.1 Introduction......Page 218
8.2 Quantum inverse scattering method......Page 219
8.2.1 Realizations of the Yang–Baxter algebra......Page 221
8.3 Algebraic Bethe ansatz method of solution......Page 222
8.3.1 Scalar products of states......Page 223
8.4 A model for two coupled Bose–Einstein condensates......Page 224
8.4.1 Asymptotic analysis of the solution......Page 226
8.5 A model for atomic–molecular Bose–Einstein condensation......Page 230
8.5.1 Asymptotic analysis of the solution......Page 233
8.5.2 Computing the energy spectrum......Page 234
8.6 The BCS Hamiltonian......Page 236
8.6.1 A universally integrable system......Page 237
8.6.2 Asymptotic analysis of the solution......Page 240
Acknowledgments......Page 241
References......Page 242
9.1 Introduction......Page 244
9.2 Lattice path integral and quantum transfer matrix......Page 246
9.2.1 Mapping to a classical model......Page 247
9.2.2 Bethe ansatz equations......Page 249
9.3.1 Derivation of nonlinear integral equations......Page 250
9.3.2 Integral expressions for the eigenvalue......Page 252
9.4 Numerical results......Page 253
9.5 Low-temperature asymptotics......Page 255
9.5.1.1 Particle–hole excitations from left to right Fermi point d  = 0 (s = 0)......Page 257
9.5.1.2 Excitations with spin s  = 0 (d = 0)......Page 258
9.5.1.3 General case: arbitrary d, s......Page 259
9.5.2 O(1/B) corrections......Page 260
9.5.4 O(1/B) corrections to the eigenvalue......Page 261
Acknowledgments......Page 262
Appendix......Page 263
References......Page 264
10.1 Reaction–diffusion processes......Page 266
10.2 Quantum Hamiltonian formulation......Page 270
10.3 Hecke algebra and integrability......Page 272
10.4 Single-species models......Page 275
10.5 The seven-vertex model......Page 280
10.6 Further applications......Page 282
10.6.2 Similarity transformations......Page 283
10.6.3 Free fermions......Page 285
10.6.4 Partial integrability......Page 286
10.6.5 Multi-species models......Page 288
10.6.6 Diffusion algebras......Page 290
10.7 Outlook: local scale-invariance......Page 291
References......Page 295