Faddeev L.D., Takhtajan L., Reyman A.G. — Hamiltonian methods in the theory of solitons :: Электронная библиотека попечительского совета мехмата МГУ

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Faddeev L.D., Takhtajan L., Reyman A.G. — Hamiltonian methods in the theory of solitons

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Название: Hamiltonian methods in the theory of solitons

Авторы: Faddeev L.D., Takhtajan L., Reyman A.G.

Аннотация:

The main characteristic of this now classic exposition of the inverse scattering method and its applications to soliton theory is its consistent Hamiltonian approach to the theory. The nonlinear Schr?dinger equation, rather than the (more usual) KdV equation, is considered as a main example. The investigation of this equation forms the first part of the book. The second part is devoted to such fundamental models as the sine-Gordon equation, Heisenberg equation, Toda lattice, etc, the classification of integrable models and the methods for constructing their solutions.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2007

Количество страниц: 597

Добавлена в каталог: 07.03.2014

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Предметный указатель

Lattice Landau — Lifshitz (LLL) model, local functionals for      512
Lattice Landau — Lifshitz (LLL) model, monodromy matrix for      511
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) model      299
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) model, auxiliary linear problem for      302
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) model, continuum limit of      301
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) model, equations of motion of      301
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) model, Hamiltonian for      301
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) model, zero curvature representation for      302
Lattice nonlinear Schr $\ddot{o}$ dinger 1 ( $LNS_{1}$ ) modelPoisson structure of      299
Lattice nonlinear Schr $\ddot{o}$ dinger 2 ( $LNS_{2}$ ) model      303
Lattice nonlinear Schr $\ddot{o}$ dinger 2 ( $LNS_{2}$ ) model, equations of motion of      303
Lattice nonlinear Schr $\ddot{o}$ dinger 2 ( $LNS_{2}$ ) model, Hamiltonian for      304
Lattice nonlinear Schr $\ddot{o}$ dinger 2 ( $LNS_{2}$ ) model, Poisson structure of      303
Lattice nonlinear Schr $\ddot{o}$ dinger 2 ( $LNS_{2}$ ) model, zero curvature representation for      304
Lattice sine — Gordon (LSG) model      516
Lattice sine — Gordon (LSG) model, Hamiltonian for      517 518
Lattice sine — Gordon (LSG) model, Poisson structure of      517
Lax representation      5
Legendre transformation      322
Lie — Poisson bracket      282-283 323 524 528-529 553
Lorentz boost generator      286 442
Maurer — Cartan cocycle      323 548
Maurer — Cartan cocycle form      312 330
Modified chiral field model      331-333
Modified chiral field model, action for      33 1
Modified chiral field model, Hamiltonian for      332
Modified chiral field model, Poisson structure of      332
Modified chiral field model, zero curvature representation for      332
Momentum      289-292
Momentum for the HM model      283 292 390
Momentum for the KdV model      309
Momentum for the LL model      292
Momentum for the NS model      16 232 262 291
Momentum for the SG model      286 292 438
Momentum for the SG model in light-cone variables      449
Monodromy matrix      24 345
Monodromy matrix for the LLL model      51 1
Monodromy matrix for the NS model      24
Monodromy matrix for the Toda model      474
Monodromy matrix, reduced      44
Monodromy matrix, reduced, finite density      62
Monodromy matrix, reduced, for the HM model      361
Monodromy matrix, reduced, for the NS model,rapidly decreasing      44
Monodromy matrix, reduced, for the SG model      399
Monodromy matrix, reduced, for the Toda model      480
Nonlinear Schr $\ddot{o}$ dinger (NS) equation      11
Nonlinear Schr $\ddot{o}$ dinger (NS) equation, higher      198
Nonlinear Schr $\ddot{o}$ dinger (NS) equation, vector      288
Nonlinear Schr $\ddot{o}$ dinger (NS) model      12
Nonlinear Schr $\ddot{o}$ dinger (NS) model, -soliton solution for      135 171
Nonlinear Schr $\ddot{o}$ dinger (NS) model, -matrix for      189
Nonlinear Schr $\ddot{o}$ dinger (NS) model, action-angle variables for      229-232 258
Nonlinear Schr $\ddot{o}$ dinger (NS) model, admissible functional for      17 18 75-76 195 206 209 233-235 239
Nonlinear Schr $\ddot{o}$ dinger (NS) model, algebra of observables for      12-13 221
Nonlinear Schr $\ddot{o}$ dinger (NS) model, auxiliary linear problem for      23
Nonlinear Schr $\ddot{o}$ dinger (NS) model, auxiliary linear problem for, reflectionless      126 165
Nonlinear Schr $\ddot{o}$ dinger (NS) model, boundary conditions for      11-12
Nonlinear Schr $\ddot{o}$ dinger (NS) model, canonical variables for      230-231
Nonlinear Schr $\ddot{o}$ dinger (NS) model, charge for      16 232 249
Nonlinear Schr $\ddot{o}$ dinger (NS) model, complete integrability of      229-233
Nonlinear Schr $\ddot{o}$ dinger (NS) model, condition () for      49
Nonlinear Schr $\ddot{o}$ dinger (NS) model, condition ( $\theta$ ) for      69 149
Nonlinear Schr $\ddot{o}$ dinger (NS) model, condition for the determination of signs for      71
Nonlinear Schr $\ddot{o}$ dinger (NS) model, conjugation problem for      114 148
Nonlinear Schr $\ddot{o}$ dinger (NS) model, continuous spectrum in      49 56
Nonlinear Schr $\ddot{o}$ dinger (NS) model, discrete spectrum in      49 65
Nonlinear Schr $\ddot{o}$ dinger (NS) model, dispersion law for      243 262
Nonlinear Schr $\ddot{o}$ dinger (NS) model, dispersion relation for      51 68
Nonlinear Schr $\ddot{o}$ dinger (NS) model, finite density case of      12
Nonlinear Schr $\ddot{o}$ dinger (NS) model, fundamental Poisson brackets for      189
Nonlinear Schr $\ddot{o}$ dinger (NS) model, gauge transformation of      315-321 391
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Gelfand — Levitan — Marchenko equation for      116-117 152-153
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Hamiltonian for      16 19 232 262
Nonlinear Schr $\ddot{o}$ dinger (NS) model, higher equations for      198 209 213
Nonlinear Schr $\ddot{o}$ dinger (NS) model, inadmissible functional for      19 75 235 260
Nonlinear Schr $\ddot{o}$ dinger (NS) model, integrals of the motion for      53 73-76 205 216
Nonlinear Schr $\ddot{o}$ dinger (NS) model, integrals of the motion for, generating function for      25 33-38 54 73
Nonlinear Schr $\ddot{o}$ dinger (NS) model, integrals of the motion for, local      33 214
Nonlinear Schr $\ddot{o}$ dinger (NS) model, integrals of the motion for, nonlocal      216
Nonlinear Schr $\ddot{o}$ dinger (NS) model, interaction of solitons in      133-136 171-172
Nonlinear Schr $\ddot{o}$ dinger (NS) model, inverse scattering problem for      108 146
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for      39 42 57-58
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, analytic properties of      42 59-60
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, asymptotic expansion for large $\textit{A}$       43 60
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, evolution equations for      52 72
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, integral equations for      44
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, integral representation for      40-41 57-58
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, involutions for      41 43 49 59-61
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Jost solutions for, Poisson brackets of      224 251
Nonlinear Schr $\ddot{o}$ dinger (NS) model, momentum for      16 232 262 291
Nonlinear Schr $\ddot{o}$ dinger (NS) model, monodromy matrix for      24
Nonlinear Schr $\ddot{o}$ dinger (NS) model, monodromy matrix for, asymptotic expansion of      29-37
Nonlinear Schr $\ddot{o}$ dinger (NS) model, monodromy matrix for, evolution equations for      29
Nonlinear Schr $\ddot{o}$ dinger (NS) model, monodromy matrix for, integral representation for      32
Nonlinear Schr $\ddot{o}$ dinger (NS) model, monodromy matrix for, reduced      44 62
Nonlinear Schr $\ddot{o}$ dinger (NS) model, monodromy matrix for, reduced, Poisson brackets of      224 253
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Poisson structure of      13-15 218-219 241 257
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Poisson structure of, annihilator for      20
Nonlinear Schr $\ddot{o}$ dinger (NS) model, recursion operator for      212
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Riccati equation for      35
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Riemann problem for      81 89 139 558
Nonlinear Schr $\ddot{o}$ dinger (NS) model, Riemann problem for, normalization of      84
Nonlinear Schr $\ddot{o}$ dinger (NS) model, scattering matrix for      83 138 246
Nonlinear Schr $\ddot{o}$ dinger (NS) model, scattering of solitons in      133-136 171-172 245 263
Nonlinear Schr $\ddot{o}$ dinger (NS) model, soliton dynamics in      133-136 171-172 263-266
Nonlinear Schr $\ddot{o}$ dinger (NS) model, soliton for      128-129 168-169
Nonlinear Schr $\ddot{o}$ dinger (NS) model, trace identities for      55 78
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for      28 45 50 62 67
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for integral representation for      33 64-65
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for, analytic properties of      46-51 63-72
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for, asymptotic expansion of      53-54 74
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for, evolution of      52-53 73
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for, involutions for      28
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for, normalization condition for      28 45 62
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition coefficients for, Poisson brackets of      225-226 253
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition matrix for      26
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition matrix for, asymptotic expansion of      29 34
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition matrix for, evolution of      29
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition matrix for, integral equations for      30
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition matrix for, involutions for      28
Nonlinear Schr $\ddot{o}$ dinger (NS) model, transition matrix for, Poisson brackets of      190
Nonlinear Schr $\ddot{o}$ dinger (NS) model, vector      288
Nonlinear Schr $\ddot{o}$ dinger (NS) model, zero curvature condition for      21-22 204 209 289
Normalization of the Riemann problem      336
Normalization of the Riemann problem for the NS model      84
Normalization of the Riemann problem for the SG model      410
Normalization of the Riemann problemfor the HM model      371
Normalization of the Riemann problemto unity      336
Normalization relation      28 45 62 361 400 481
Normed ring $\mathfrak{R}^{(2\times 2)}$ , $\mathfrak{R}^{(2\times 2)}_{\pm}$       85-86
Normed ring $\mathfrak{R}_{0}$ , $\mathfrak{R}_{\pm}$       48
Novikov’s equations      269
Number of particles      16
Parallel transport      22
Partially anisotropic lattice Heisenberg model      516
Pauli matrices      21
Phase space $\textit{M}_{0}$       12 237
Phase space $\textit{M}_{c}$       499
Phase space $\textit{M}_{L,\theta}$       17
Phase space $\textit{M}_{\varrho,\theta}$       18
Phase space, extended      176
Poisson brackets      13
Poisson brackets compatible      563
Poisson brackets for the $LNS_{1}$ model      299
Poisson brackets for the $LNS_{2}$ model      303
Poisson brackets for the chiral field model      322
Poisson brackets for the chiral field model, modified      332
Poisson brackets for the HM model      282
Poisson brackets for the KdV model      308
Poisson brackets for the LHM model      296

Poisson brackets for the LL model      458
Poisson brackets for the LSG model      517
Poisson brackets for the NS model      13
Poisson brackets for the NS model, vector      288
Poisson brackets for the SG model      285 431
Poisson brackets for the SG model in light-cone variables      448
Poisson brackets for the Toda model      294 471 520
Poisson brackets for the Toda model, two-dimensional      313
Poisson brackets of transition coefficients for the HM model      387-388
Poisson brackets of transition coefficients for the NS model, finite density      253
Poisson brackets of transition coefficients for the NS model, rapidly decreasing      225-226
Poisson brackets of transition coefficients for the SG model      434-435
Poisson brackets of transition coefficients for the Toda model      503
Poisson structure      13 523-524
Poisson structure, nondegenerate      15
Poisson — Dirac bracket      273 563
Pole divisor      334
Quasi-periodicity conditions      12 282 289 294
Recursion operator      212
Recursion relations      35 365 405 488
Reduction      307 310 312 314
Reflection coefficients      83 138
Reflectionless function      127
Regularization of a Fredholm equation      120 125
Riccati equation      35 365 403
Riemann problem      81 335 552
Riemann problem for the HM model      370
Riemann problem for the NS model, rapidly decreasing      89 558
Riemann problem for the NS model, rapidly decreasing, finite density      139
Riemann problem for the SG model      407
Riemann problem on a Riemann surface      139
Riemann problem with zeros      91 337
Riemann problem, regular      90
Riemann problem, scalar      99 248
Scattering matrix      83 138
Schr $\ddot{o}$ dinger operator      79
Sine — Gordon (SG) equation      284
Sine — Gordon (SG) equation, higher      437
Sine — Gordon (SG) model      284
Sine — Gordon (SG) model , reduced monodromy matrix for      399
Sine — Gordon (SG) model , reduced monodromy matrix for, asymptotic expansion of      405
Sine — Gordon (SG) model , reduced monodromy matrix for, evolution of      402
Sine — Gordon (SG) model , reduced monodromy matrix for, involutions for      399
Sine — Gordon (SG) model , reduced monodromy matrix for, Poisson brackets of      434
Sine — Gordon (SG) model, -soliton solution for      429
Sine — Gordon (SG) model, -matrix for      433
Sine — Gordon (SG) model, action-angle variables for      435-437
Sine — Gordon (SG) model, antisoliton for      427
Sine — Gordon (SG) model, auxiliary linear problem for      393 450
Sine — Gordon (SG) model, boost generator for      286 442
Sine — Gordon (SG) model, boundary conditions for      284-285
Sine — Gordon (SG) model, breather for      428
Sine — Gordon (SG) model, complete integrability of      435-437
Sine — Gordon (SG) model, condition () for      400
Sine — Gordon (SG) model, discrete spectrum in      401
Sine — Gordon (SG) model, dispersion law for      438
Sine — Gordon (SG) model, dispersion relations for      401
Sine — Gordon (SG) model, double soliton for      428
Sine — Gordon (SG) model, excitation spectrum in      439
Sine — Gordon (SG) model, fundamental Poisson brackets for      433
Sine — Gordon (SG) model, Gelfand — Levitan — Marchenko equation for      416-417
Sine — Gordon (SG) model, Hamiltonian for      285 406 438 448
Sine — Gordon (SG) model, higher equations for      436 437
Sine — Gordon (SG) model, in light-cone variables      446
Sine — Gordon (SG) model, integrals of the motion for      403-407 452
Sine — Gordon (SG) model, integrals of the motion for, generating function for      406
Sine — Gordon (SG) model, inverse scattering problem for      407
Sine — Gordon (SG) model, Jost solutions for      396 453
Sine — Gordon (SG) model, Jost solutions for, analytic properties of      398
Sine — Gordon (SG) model, Jost solutions for, asymptotic behaviour of      396 398-399
Sine — Gordon (SG) model, Jost solutions for, evolution equations for      402
Sine — Gordon (SG) model, Jost solutions for, integral representation of      397
Sine — Gordon (SG) model, Jost solutions for, involutions for      396 399
Sine — Gordon (SG) model, Jost solutions for, Poisson brackets of      434
Sine — Gordon (SG) model, momentum for      286 292 406 438 449
Sine — Gordon (SG) model, Poincar $\acute{e}$ algebra action for      439-442
Sine — Gordon (SG) model, Poisson structure of      285 431 448
Sine — Gordon (SG) model, Riccati equation for      403
Sine — Gordon (SG) model, Riemann problem for      407
Sine — Gordon (SG) model, Riemann problem for, normalization of      410
Sine — Gordon (SG) model, scattering of solitons in      442-446
Sine — Gordon (SG) model, scattering transformation for      429-431 444
Sine — Gordon (SG) model, soliton for      427
Sine — Gordon (SG) model, topological charge for      285
Sine — Gordon (SG) model, trace identities for      407
Sine — Gordon (SG) model, transition coefficients for      400-401
Sine — Gordon (SG) model, transition coefficients for, analytic properties of      400
Sine — Gordon (SG) model, transition coefficients for, asymptotic expansion of      406 452
Sine — Gordon (SG) model, transition coefficients for, evolution of      403 452
Sine — Gordon (SG) model, transition coefficients for, involutions for      400
Sine — Gordon (SG) model, transition coefficients for, Poisson brackets of      434-435
Sine — Gordon (SG) model, transition matrix for      394
Sine — Gordon (SG) model, transition matrix for, asymptotic expansion of      403
Sine — Gordon (SG) model, transition matrix for, evolution of      402
Sine — Gordon (SG) model, transition matrix for, involutions for      394
Sine — Gordon (SG) model, transition matrix for, Poisson brackets of      433
Sine — Gordon (SG) model, zero curvature representation for      286 450
Sine — Gordon matrix model      349
Sochocki — Plemelj formula      50 501
Soliton      128
Soliton dynamics for the HM model      383
Soliton dynamics for the NS model, finite density      171
Soliton dynamics for the NS model, rapidly decreasing      133
Soliton dynamics for the SG model      429-430
Soliton dynamics for the Toda model      498
Soliton for the HM model      379-381
Soliton for the NS model, rapidly decreasing      128-129
Soliton for the NS model, rapidly decreasing, finite density      168-169
Soliton for the SG model      427
Soliton for the Toda model      496
Spin      283 368
Superposition formula      23
Sympletic structure      15
Sympletic structure form      15 291-292
Toda model      293
Toda model, -soliton solution for      497-498
Toda model, -matrix for      473
Toda model, action-angle variables for      503 506
Toda model, admissible functionals for      499 505 506
Toda model, auxiliary linear problem for      294 475
Toda model, boundary conditions for      293-294
Toda model, complete integrability of      471 499
Toda model, condition for the determination of signs for      485
Toda model, continuous spectrum in      477
Toda model, discrete spectrum in      483
Toda model, dispersion relations for      484
Toda model, equations of motion of      293
Toda model, fundamental Poisson brackets for      472
Toda model, gauge transformation for      478
Toda model, Gelfand — Levitan — Marchenko equation for      491
Toda model, Hamiltonian for      294 472 504
Toda model, higher equations for      506
Toda model, inadmissible functional for      499 505 506
Toda model, integrals of the motion for      474 487
Toda model, integrals of the motion for, local      474 487
Toda model, integrals of the motion for, local, generating function for      487
Toda model, integrals of the motion for, local, recursion relations for      488
Toda model, interaction of solitons in      498
Toda model, inverse scattering problem for      489
Toda model, Jost solutions for      477
Toda model, Jost solutions for, analytic properties of      478
Toda model, Jost solutions for, asymptotic behaviour of      478
Toda model, Jost solutions for, evolution equations for      486
Toda model, Jost solutions for, involutions for      478
Toda model, Jost solutions for, Poisson brackets of      500
Toda model, monodromy matrix for      474
Toda model, monodromy matrix for, reduced      480
Toda model, monodromy matrix for, reduced, evolution equation for      487
Toda model, monodromy matrix for, reduced, involution for      481
Toda model, monodromy matrix for, reduced, Poisson brackets of      500

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