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Conte R. — The Painlevé property: One century later
Conte R. — The Painlevé property: One century later



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Название: The Painlevé property: One century later

Автор: Conte R.

Аннотация:

The subject this volume is explicit integration, that is, the analytical as opposed to the numerical solution, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). Such equations describe many physical phenomena, their analytic solutions (particular solutions, first integral, and so forth) are in many cases preferable to numerical computation, which may be long, costly and, worst, subject to numerical errors. In addition, the analytic approach can provide a global knowledge of the solution, while the numerical approach is always local. Explicit integration is based on the powerful methods based on an in-depth study of singularities, that were first used by Poincaré and subsequently developed by Painlevé in his famous Leçons de Stockholm of 1895. The recent interest in the subject and in the equations investigated by Painlevé dates back about thirty years ago, arising from three, apparently disjoint, fields: the Ising model of statistical physics and field theory, propagation of solitons, and dynamical systems. The chapters in this volume, based on courses given at Cargèse 1998, alternate mathematics and physics; they are intended to bring researchers entering the field to the level of present research.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 810

Добавлена в каталог: 06.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A_{1}$-system      773
$A_{2}$-system      773
$\alpha$-index      121
$\psi$-series      88 111
$\tau$ function      45 342 739 741
$\tau$ function of the KdV hierarchy      272 278
Abelian subalgebras      616
Abstract Lie algebra      604
Airy equation      192 275
Airy function      576
AKNS equation      540
AKNS scheme      527 530 547 554
AKNS system      552—554 561
Algorithm      592
Allowed transformations      598 599
Alternative discrete Gambier      34 453
Alternative discrete P1 equation      495 496
Alternative discrete P2 equation      454 470 498
Analytic equivalence      347
Anti-Stokes line      195
Apparent singularity      72 770
Asymptotic expansion      352
Asymptotic series      186 188
Asymptotics      789
Asymptotics beyond all orders      188
Asymptotology      789
Auto-Baecklund transformation      469 482
Baecklund gauge transformation      529
Baecklund transformation      168 518 519 524 525 527 529—533 535 540—543 545 546 550 552 553 694
Baker — Campbell — Hausdorff formula      608
Bessel equation      194 198
Bilinear form      767
Bilinear method of Hirota      527 539 544 642 661
Binomial equations      105
Birational transformation      102 450
Blowing up      749
boundary conditions      611
Boussinesq equation      624 627—629 634 639 641
Braid group      363 393
Branching      791
Broer — Kaup equation      554
Burgers equation      519 525 538 555
Canonical basis      604
Canonical coordinates      337
Cartan equivalence classes      102
Casorati determinant      482
Characteristic equation      15 19
Charge      290 291
Chazy equation      578
Classical      639
Classical function      754
Classical solution      762
Classification      605 607
Classification of the subalgebras      615
Cnoidal waves      609
Commutation relations      607
Complete equation      122
Complex Ginzburg — Landau (CGL3) equation      556 561
Conditional symmetries      623 645
Conformal field theory      232
Connection formula      576
Connection matrix      40 49 356
Connection problem      189
Conservation law      523
Contiguity relation      763 779
continuum limit      434 448
Contravariant metric      381
Convergence of a series      188
Coxeter group      379 398
Coxeter number      399
Cremona transformation      419
Critical      95—97
Crude Stokes phenomenon      186
Crum transformation      528
D'Alembert equation      525
Darboux transformation (DT)      527—529 531 533 535 540 543 544
Darboux — Halphen system      96 144
Deautonomization      466 492
Deformation parameters      60
Degenerate solution      202
Degree      103
Determining equations      593 595 597 598 601 602 623 625 633 645
Differential field      753
Differential Groebner bases      644
Differential-difference equation      448
Diffgrob2      645
Dilatational invariance      625
Dilatations      598 603 611
DIRECT      639
Direct method      624 626 635 639 640 643—645
Direct reductions      623
Direct sum      616
Discrete bilinear form      472
Discrete birational transformation      451
Discrete Gambier      34 450
Discrete Lax pair      430
Discrete Miura transformation      451
Discrete P1 equation      248 434 455 465 494
Discrete P2 equation      449 458 496
Discrete P3 equation      494 498 500 501
Discrete P4 equation      502
Discrete Painleve equations      682
Discrete Painleve property      150
Discrete Painleve test      150 433 456
Discrete Riccati      480
Discrete Schwarzian      445
Dominant      56
Dominant balance      189 206 793
Dominant terms      85
Dynkin diagram      329 751 753
Eckhaus equation      519 538
Elliptic equation      79 207 208
Elliptic function      83 215 520 523 560 561 609 613 626 629—631 634 640
Elliptic integral      80
Entire function      83 661 662
Equations of associativity      292
Essential singularity      98 104
Expansion variable $\chi$      156
Fatgraph      235
Feynman diagrams      236 271
Fiber-preserving transformations      599 643
Fitzhugh — Nagumo equation      558 562 639 642
Fixed singularity      81 165
Flat coordinates      306 329 382
Flatness      306
Forced Korteweg — de Vries      596
Frobenius algebra      289
Frobenius manifold      306 309
Frobenius supermanifold      309
Fuchs indices      87 111 139
Fuchsian      36
Fuchsian singularity      110 111
Function      80 97
Fundamental set      4
Galilei      603
Galilei group      611
Galilei transformations      598
Garnier system      771
Gauge invariance      666
Gauss — Manin system      382
Generalized Darboux — Halphen system      577
Generalized KdV flows of D-type      269
Generalized KdV hierarchy      265
Generalized P1 equations      258
Generic solution      199
Goursat method      619
Graded Frobenius algebra      290
Grading operator      290
Gravity      288
Gromov — Witten invariant      295 297 342
Group invariant solutions      604
Hamiltonian vector field      753
Hard-dimer model      243
Heisenberg algebra      607
Henon — Heiles system      151
Higher-order NLS equation (HNLS)      520
Hirota — Satsuma equation      541
Hirzebruch surface      746
Hodograph transformation      102
Holonomic deformation      770—772
Homographic group      101
Implicit solutions      606
Index—1      86 126 138
Indicial equation      86 111
Instanton      294
Integrability      416 418 476
Integrable equation      596 601 603 607
Integrate      97
Integrates tronquees      202
Integrating Painleve equations      672
Intersection numbers      270
Invariant divisor      754
Invariant surface condition      630 632—634 644
Invariants      68 605
Inverse scattering      624
Inverse spectral transform      80 574 596 601 607
Irreducibility      79 103
Irreducible function      755
Irreducibly embedded      617 618
Ising model      230 261 267
Isomonodromic deformation problem      37
Isomonodromy problem      184
Itzykson — Zuber integral      261
Jacobi ring      292
Jacobian      605 606
Kac — Moody — Virasoro algebra      603
Kac — van Moerbeke — Volterra equation      449
Kadomtsev — Petviashvili equation      596 639
Kadomtsev — Petviashvili flows      267
Kaup — Kupershmidt equation      534 535 545
Korteweg — de Vries equation      518 520 523 527 528 530 532 539 540 547 551 554 596 607 623
Korteweg — de Vries flows      260
Korteweg — de Vries hierarchy      259
Kowalevski exponents      139
KPP equation      558 562 639 642
Kuramoto — Sivashinsky (KS) equation      559
Lagrangian correspondence      426
Lame equation      631
Landau — Ginzburg model      374
Lattice equation      443
Lattice KdV      445 489
Lattice MKdV      490
Laurent series      77 83 84
Lax pair      165 528
Lax pair of second order      535 537 540 550 560 562
Lax pair of third order      535 541 543 545 559 562
Levi decomposition      604
Lie algebra      594
Linear group characters      276
Linearized Equation      87 114 115
Liouville equation      525 526 538
Loop equations      280
Lorenz model      78 123 154 163
Macsyma      625 633 634
Matrix integral      234 272
Matter sector      289
Maximal      617
Meromorphy assumption      83
Meromorphy test      84
Method of pole-like expansions      118
Mirror conjecture      298 332
Miura transformation      450 541 542 544 547
Modified Kadomtsev — Petviashvili      601
Modified Korteweg — de Vries equation      183 520 524 527 531 547 550 552 554 575
Moduli space of Riemann surfaces      270
monodromy      486 576 585
Monodromy data      40 321 324 345
Monodromy group      386
Monodromy matrices      37
Monodromy representation      36
Movable singularity      81
Multiple matrix integrals      263
Nahm system      579
Nearly degenerate solution      201
Newton's polygon      86
Nilpotent      604
Nilradical      604 607
Non-Fuchsian singularity      110 113 145
Nonclassical method      624 632 635 639 645
Nonlinear Schroedinger equation      520 522 524 527 533 547 554 611 613
Nontwisted subalgebras      619
Normal form      317
One-parameter solutions of Pn      696 705 709 712 721 728 729 733
Orthogonal polynomials      246
P1 equation      205 251 252 626 628 629 667 673 795
P1 transcendent      610
P2 equation      205 575 626 628 629 668 674 695
P2 transcendent      610
P3 equation      670 675 719
P4 equation      626 628 629 677 708
P5 equation      678 729
P6 equation      305 383 385 386 679 732
Painleve analysis      591 607
Painleve property      80 81 95 100 182 573 691 791
Painleve property for PDEs      518
Painleve test      95 108 153 791
Painleve test for PDEs      518 529 535—538 554—556 558
Painleve — Darboux equations      540
Parabolic subalgebra      619
Partial monodromy data      41
Partially invariant solutions      606 613
Pencil      381
Period map      332 386
Perturbation of the continuum limit      150
Poincare rank      52
Point transformations      592
PolyPainleve test      798
Primary chiral algebra      290
Primary chiral field      289
Principal part of a Laurent series      83
Projective Riccati system      541 543 545
Prolongation      632
Prolongation of the vector field      594 595
Prolongation theory      593
Pseudorthogonal Lie algebra      618
Puiseux diagram      86
Quadratic map      420 749
Quadratures      80
Quantum cohomology      295
Quantum field theory      288
Quantum gravity      229
Random surfaces      234
Rational dependence      83
Rational solutions of Pn      700 701 707 716 719 722 723 727 731
Recessive      56
Reducibly embedded      617
Reducibly embedded subalgebras      618
Reduction      183 574
Resolvent      239
Resonances      139 795
Reversible mapping      424
Riccati equation      77 97 102 103 155 530—532 537 539 546 548—550 553 558 560
Riemann — Hilbert problem      37
Saddle-point approximation      240 245
Sawada — Kotera equation      534 535 541 544
Schlesinger transformation      348 469 482
Schwarzian derivative      95 102
Schwarzian equation      155 587
Self-dual Yang — Mills      578
Semidirect sum      616 622
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