PARTICIPANTS

ABLOWITZ MARK J., Program in Applied Mathematics, University of Colorado at Boulder, Campus Box 526, Boulder Colorado 80309-0526, USA.

BEATTY MORRIS, Institute for Theoretical Dynamics, University of California at Davis, Davis, CA 95616, USA.

BUREAU F.J., Place d'Italie 5(042), B-4020 Liege, Belgium.

CLARKE FRANCIS, Directeur, Centre de recherches mathematiques, Universite de Montreal, C.P. 6128-A, Montreal, Quebec, Canada, H3C 3J7.

CLARKSON PETER A., Department of Mathematics, Exeter University, Exeter, EX4 QE, England.

CONTE ROBERT, SPSRM, CEN Saclay, F 91191 Gif-sur-Yvette Cedex, France.

ERCOLANI N.M., Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA.

FOKAS A.S., Department of Math. and Comp. Sc., Clarkson University, Potsdam, NY 13676, USA.

GAGNON LANGIS, Departement de physique, Universite Laval, Ste-Foy, Quebec, Canada, G1K 7P4.

GOLDSTEIN PETER, Soltan Institute for Nuclear Studies, 05-400 Otwock­Swierky, Poland.

GRAMMATICOS BASILE , LPN, Universite Paris VII, Tour 24-14, 5ieme etage, 2, Place Jussieu, 75251 Paris, France.

GRUNDLAND MICHEL, Centre de recherches matMmatiques, Universite de Montreal, C.P. 6128-A, Montreal, Quebec, Canada, H3C 3J7.

HIETARINTA JARMO, Department of Physical Sciences, University of Turku, SF-20500 Turku 50, Finland.

HLAVATY L., Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 18040, Prague 8, Czechoslovakia.

433

HURTUBISE JACQUES, Department of Mathematics, McGill University, Montreal, Quebec, Canada, H3A 2K6.

RUSSIN VERONIQUE, Centre de recherches mathematiques, Universite de Montreal, C.P.6128-A, Montreal, Quebec, Canada, H3C 3J7.

JOSHI NALINI, Department of Mathematics, University of New South Wales, Sydney, NSW, Australia 2033.

KAMRAN NIKY, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke ouest, Montreal, Quebec, Canada, H3A 2K6.

KAUP DAVID, Department of Physics, Clarkson University, Potsdam, NY 13676, USA.

KITAEV ALEXANDER, Department of Mathematics, Leningrad Institute of Air­craft Engineering, 805 Gerzen 67, Leningrad, 190000, USSR.

KRUSKAL M., Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA.

LAW CHUNKONG, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.

LEGARE MARTIN, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada, T6G 2Jl.

LEVI DEClO, Dipartimento di Fisica, Universita di Roma La Sapienza, Piazzale A. Mora 2, I-00185 Roma, Italy.

LU YOUMIN, c/o Dr. J.B. McLeod, Department of Mathematics, University of Pittsburgh, NY 15260, USA.

McCOY BARRY, Department of Physics, SUNY, Stony Brook, NY 11794-3840, USA.

McLEOD J. BRYCE, Department of Mathematics, University of Pittsburgh, Pitts­burgh, PA 15260, USA.

MELKONIAN SAM, Department of Mathematics, Carleton University, Ottawa, Ontario, Canada, K1S 5B6.

MENYUK CURTIS, Department of Elect. Engineering, University of Maryland, Baltimore County Campus, Baltimore, MD 21228, USA.

MUGAN UGURHAN, Department of Mathematics, University Bilkent, P.O. 8, 06572 Maltepe, Ankara, Turkey.

MUSETTE MICHELINE, V.U.B., Dienst TENA V.V.B., Pleinlaan 2, 1050 Brussel, Belgium.

434

OKAMOTO KAZUO, College of Arts and Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan.

PAPAGEORGIOU VASSILIS, Dept. of Math. and Comp. Sc., Clarkson University, Potsdam, NY 13676, USA.

SCHIFF JEREMY, Department of Physics, Columbia University, New York, NY 10027, USA.

SINGER STEPHANIE, Courant Institute, 251 Mercer St., New York, NY 10012, USA.

TABOR MICHAEL, Department of Applied Physics, Columbia University, New York, NY 10027, USA.

TAMIZHMANI K.M., Department of Mathematics, Pondicherry University, Kalapet, Pondicherry 605104, India.

TRACY CRAIG, Department of Mathematics, University of California at Davis, Davis, CA 95616, USA.

WEISS JOHN, 49 Grandview Road, Arlington, Massachusetts, MA 02174, USA.

WINTERNITZ PAVEL, Centre de recherches mathematiques, Universite de Montreal, C.P. 6128-A, Montreal, Quebec, Canada, H3C 3J7.

435

PARTICIPANTS

1. Y. Lu 18 . B. McCoy 2 . D. Levi 19 . R. Conte 3 . s . Chenevert 20 . v. Hussi n 4 . N. Joshi 21. L. Lalague 5. c . Law 22 . s . Melkoni an 6 . P. Goldstein 23 . M. Legare 7. P. Winternitz 24 . M. Ablowitz 8 . M. Musette 25 . J . Weiss 9 . F. J . Bureau 26 . M. Tabor

10 . M. Winternitz 27 . J . Schiff 11 . B. Gr amrnaticos 28 . J . Hietarinta 12 . M. Grundland 29 . L. Hlavaty 13 . K.M. Tamizhmani 30 . u. Mug an 14 . J . B. McLeod 31. P. A. Clarkson 15 . A. Kitaev 32 . C.A. Tracy 16 . M. Kruskal 33 . D.J . Kaup 17 . K. Okamoto

436

AUTHOR INDEX

Ablowitz, M.J., 1-8, 13, 17, 20, 29, 34, 38, 50, 62, 81, 82, 93, 97, 98, 125, 128, 131, 145, 148, 168, 198, 206, 206, 211, 218, 220, 226, 261, 262, 264, 267, 268, 331, 332, 334, 339, 342, 349, 350, 375, 386, 397, 399, 410, 426, 429

Abramowitz, M., 16-17 Adler M.,147, 260, 339 Ahlfors L.V., 64 Airault H., 29, 396 Albertini G., 387, 389 Arnold V.I., 298, 338 Atiyah M.F., 399, 412 Au-Yang H., 387, 388, 409

Bariev R.Z., 409 BarouchE., 377, 380, 407, 409 Barth W., 298 Bassom A.P., 4 Bateman H., 85-87 Baxter R.J., 379 Bazhanov V.V., 388 Belavin A.A.,377, 387, 399 Belinskii V.A., 429 Bender C.M., 4, 63, 65, 71, 72,

190, 357 Benney D.J., 427 Bianchi L., 280 Bieberbach L., 104, 106 Birkhoff G.D., 39 Bloembergen N., 362, 365 Bluman G.W., 211, 212, 321, 326,

327 Bobenko A.I., 339 Boiti M., 29, 426, 427 Bordag L.A., 82, 92 Bountis T., 123, 145, 218 Boutroux P., 4, 62, 63, 356 Brezin E., 36, 409 Bruschi M., 211, 214 Buchholz H., 17 Bureau F.J., 106, 107, 113, 115, 116,

125, 127, 129, 141, 143, 146, 266, 312, 425, 429, 430

Burnat M., 163, 169 Burtsev S.P., 429

Callan C.G., 409, 415 Calogero F., 198, 200 Can M., 5 Capel H.W., 377, 384, 410 Cariello F., 320-325

Carlsten J.L., 364 Carman R.L., 362, 365 Carnevale G., 125, 128, 131, 137, 197,

218, 226, 261, 262, 268, 318 Cartan E., 165, 271, 273, 274 Caudrey P.J., 3 Chadan K., 198, 200 Chakravarty S., 331, 339, 342, 399 Champagne B., 215 Chang Y.F., 239, 317 Chau L.L., 399, 401 Chazy J., 332, 340 Chowdhury A.R., 5 Chu F.Y.F., 345, 354, 364 Chudnovsky D.V., 319 Chudnovsky G.V., 319 Churchill R.C., 151 Clarkson P.A., 3, 6, 17, 19, 29, 56,

159, 187, 211, 229, 331, 339, 342, 399

Cole J.D., 321, 326, 327 Conte R., 125, 129, 132, 133, 137,

139, 197, 204, 207, 218, 320 Corliss G., 239 Cornille H., 428 Corrigan E., 399 Cosgrove C.M., 17, 19 Courant R., 228 Creamer D., 62, 409

Darboux G., 137, 163, 198, 200, 232, 244, 245

Das A., 399 Date E., 388 Davey R., 408, 409, 420 David D., 211 Davis P., 237, 255 de Boer P.C.T., 4 Degasperis A., 198, 200 Deift P., 3, 232 Deser S., 393, 394 Dolan L., 399, 401 Dorizzi B., 148 Dotsenko H.S., 387 Douglas M.R., 36, 409 Drazin P.G., 402 Drinfeld V.G., 377, 399 Drouffe J.M., 139 Dri.ihl K., 364 Duncan M.D., 354, 362-366

Ercolani N., 147, 218, 249, 252, 253, 257, 260

Erdelyi A., 16, 54, 85-87 Erugin N.P., 97

437

Fackerell E.D., 4 Fairlie D.B., 399 Fedoruk M.V., 94 Flaschka H., 5, 34, 40, 49, 56, 97,

249, 251-253, 257, 260, 319 Florjanczyk M., 3 Fokas A. S., 2, 5, 29, 34-37, 41,

50, 56, 81, 82, 97, 98, 212, 402, 404, 410

Fordy A., 139, 205 Fournier J.D., 316, 317, 320, 325 Fuch J.C., 211 Fuchs L., 312, 410 Fuchs R., 5, 312, 410 Fuchssteiner B., 212, 402, 404

Gagnon L., 3, 426 Gambier B., 97, 141, 175, 312, 425,

430 Garcet A., 123 Gardner C.S., 2 Gardner R.B., 165, 271, 274 Garnier R., 5, 62, 312 Giannini J.A., 4 Gibbon J.D., 29, 218, 319, 402 Gibbons G.W., 340 Gibbons J., 139, 205 Goddard P., 399 Goffar J., 123 Goodman R., 250, 251 Goursat E., 163 Grarrunaticos B., 123, 145, 151, 153,

155, 157, 160, 187, 215, 218, 426

Greene J.M., 2, 317 Gromak V.I., 29, 97 Gross D.J., 36, 62, 409 Grundland A.M., 163, 169, 170, 326 Gunning R.C., 271, 294

Haberman R., 3-4 Raine L., 249, 253, 257, 260 Hall P., 4 Hastings S.P., 2, 4, 5, 7, 8, 50, 62 Hawking S.W., 340 Hearn A.C., 215 Helgason S., 412 Hereman W., 228 Hietarinta J., 155, 176, 178, 179,

208, 402 Hilbert D., 228 Hilfer G., 346 Hille E.E., 2, 4, 227, 231 Hirota R., 139, 176, 198, 207, 207 Hitchin N.J., 272, 284, 399 Hlavaty L., 211, 215, 261, 263, 264,

266, 267, 269 Hong J., 394, 395, 399 Hoyer P., 105, 114, 125 Hsu L., 271-273, 275, 279, 280 Hukuhara M., 312 Hurtubise J., 272, 284, 285

Ince E.L., 2, 4, 82, 97, 141, 146, 218, 226, 261, 266, 275, 279, 294, 296, 297' 356, 425, 430

Ishii M., 152 Ito H., 159 Ito M., 208 Its A. R., 5, 13, 18, 35, 37, 50,

53-56, 81, 84-86, 384, 409 Itzykson C., 407 Izergin A.G., 384, 409

438

Jackiw R., 393-395, 399 Jimbo M., 5, 34, 49, 62, 81, 84, 90,

92, 97' 134, 377' 378, 380, 381, 387, 399, 407, 409, 410, 413-415

John F., 168 Johnson R.S., 402 Joseph R.I., 4 Joshi N., 5, 56, 62, 63, 65, 68, 69,

179, 191

Kac M., 34 Kaliappan P., 211 Kamran N., 165, 211, 271-273, 275,

276, 279, 280, 284, 285 Kapaev A.A., 5, 50, 55, 56, 62, 81,

85, 86 Kashaev R.M., 388 Kashiwara M., 415 Kaup D.J., 2, 3, 6, 17, 18, 97,

198, 206, 206, 345-350, 354, 364, 370, 371, 427-429

Kawamoto S., 211 Kazakov V.A., 409 Khan T.M., 5 Kim Y., 394, 395, 399 Kimura H., 312, 313 Kitaev A.A., 5, 35, 50, 56, 81-83,

85-87' 90-92 Kong X.P., 409 Konno K., 200 Konstant B., 250, 251 Korepin V.E., 384, 409 Kosmann-Schwarzbach Y., 211 Kovalevsky S., 188, 226 Krasil'shchik I.S., 211, 212 Krasnosel'ski1 M.A., 71 Krichever I.M., 56, 399 Krishnan K.S., 361 Kruskal M.D., 2, 3, 5, 56, 62, 63,

65, 68, 69, 129, 134, 145, 159, 160, 179, 187, 190, 191, 195, 211, 218, 399

Kumei S., 211, 212 Kurruner M., 151

Lakin W.D., 4 Lakshmanan M., 211, 212 Lamb K., 271, 273, 275, 276, 279 Landsberg G., 361 Lavie M., 126, 231 Lebeau G., 5, 6 LeeK., 394, 395, 399 Leon J.J.P., 427 Levi D., 137, 198, 200, 201, 211,

214, 354, 357, 360, 372, 428

Levine G., 123, 316-318, 325 Leznov A.N., 399 Li Yi-Shen., 212 Lochak P., 5, 6 Lu J.H., 253, 282 Ludford G.S.S., 4 Lukashevich N.A., 29, 97 Luke J.C., 427 Lychagin V.V., 211, 212

Mack M.E., 362 Magnus W., 16, 101 Magri F., 212 Mahon R., 354, 362-366 Malgrange B., 413 Malmquist J., 313

Manakov S.V., 36, 333, 335, 338, 346, 349

Mandelstam L., 361 Mangazeev V.V., 388 Manin Yu.I., 399 Marcilhacy G., 4 Marinari E., 36 Martina L., 428 Mason L.J., 331, 335, 399, 402, 404 Matsukidaira J., 402 McCaskill J.S., 4 McCoy B.M., 4, 5, 13, 50, 62, 81, 377,

379-382, 385-389, 407, 409, 410

McKean H.P.Jr., 249, 253 McLeod J.B., 2, 4-8, 50, 56, 62, 211,

218, 226 Meinel R., 211, 214 Menyuk C.R., 346, 349, 350, 354, 357,

362, 364, 365, 367-372 Migdal A.A., 36, 62, 409 Mikhailov A.V., 429 Miki K., 388 Miles J.W., 4 Miura R.M., 2 Miwa T., 5, 34, 49, 62, 84, 90, 92,

97, 134, 377, 378, 380, 381, 387, 399, 407, 409, 410, 413-415

Moore G., 91, 409 Mori Y., 62, 409 Moser J., 253 Mugan U., 5, 29, 34, 37, 81, 97, 98 MUller G., 409 Munford D., 297 Murato Y., 29, 313 Musette M., 129, 133, 137, 197, 200,

204, 207, 320 Myers J.M., 420

Nahm W., 331, 339 Nakamura A., 179 Narayanan R., 408, 409, 411, 412,

414, 415, 420 Naskar M., 5 Nehari z., 341, 342 Neugebauer G., 211, 214 Newell A.C., 2, 3, 5, 6, 17, 18, 29,

34, 40, 49, 56, 97, 132, 198, 206, 206, 218, 319, 320, 349, 350, 429

Ng W.K., 362 Nijhoff F.W., 377, 384, 410 Nishioka K., 313 Novikov S.P., 56, 335, 399 Novokshenov V.Yu., 5, 13, 18, 37, 50,

53, 54, 56, 81, 84, 346, 349, 384

Nucci M.C., 218

Oberhettinger F., 16, 101 Oevel w., 212 Okamoto K., 5, 29, 94, 175, 176,

298, 312, 313 Olver P.J., 2, 211, 212, 218, 226 Onsager L., 387, 389 Orlov Yu.A., 92 Orszag S.A., 4, 63, 65, 71, 72,

190, 357 Osgood W.F., 226 Ovsiannikov L.V., 211

Pac P.Y., 394, 395, 399 Painleve P., 61, 97, 105, 126-128,

141, 175, 183, 187, 188, 275, 313, 425, 430

Palmer J., 407-409, 415, 416, 418-420

Paquin G., 427 Parisi G., 36 Pauling L .. , 17 Pempinelli F., 29, 426, 427 Peradzynski z., 163, 169, 170 Perk J.H.H., 62, 377, 381, 383,

384, 386-389, 409, 410 Peters C., 298 Petrov V.E., 50 Pilloni L., 428 Pinney E., 141 Pitayevski L.P., 335 Plemelj J., 156 Polyakov A.M., 377, 387 Pope C.N., 340 Punithavathy P., 211, 215

Quillen D., 418 Quispel G.R.W., 377, 384, 410

Radmore P., 218, 319, 402 Ragnisco 0., 211, 214 Raman C.V., 361 Ramani A., 2, 4, 8, 13, 17, 20, 62,

97, 123, 125, 128, 131, 145, 148, 151, 153, 155, 157, 160, 168, 187, 211, 218, 220, 226, 261, 262, 264, 267, 268, 397, 426

Reid G.J., 211 Reitjes J.F., 354, 362-366 Reyman A.G., 339 Richter E.W., 211 Riemann B., 163 Rittenberg v., 389 Rod D.L., 151 Rosales R.R., 4 Roseneau P., 215 Roy s., 399 Ruijsenaars S.N.N., 408

Sabatier P.C., 198, 200 Sachdev P.L., 211 Sah C.H., 387, 388 Saitoh N., 179 Salerno M., 211,215 Saleur H., 407 Santini P.M., 212, 402, 428 Sanuki H., 200 Sato M., 62, 377, 380, 387, 407, 409,

410 Satsuma J., 139, 198, 207, 402 Saveliev M.V., 399 Schwarz F., 215 Schwarz W., 249 Schwarzmeier J.L., 215 Schiff J., 394, 399 Schiff L.I., 17 Schlesinger L., 5, 313, 410 Schonfeld J., 393 Schulman E.I., 92 Scott A.C., 345, 354, 364 Segal G., 249, 416, 418

439

Segur H., 1-8, 13, 17, 20, 50, 62, 93, 97, 125, 128, 131, 145, 148, 168, 198, 206, 206, 211, 218, 220, 226, 261, 262, 264, 267, 268, 332, 334, 349, 350, 375, 386, 397, 426, 429

Seidman T.I., 364, 367, 369, 370 Semenov-Tyanshanskii M.S., 253, 339 Series C., 409 Shabat A.B., 3, 269, 348, 428 Shadwick W.F., 271, 273, 275, 276,

279, 280 Shenker S., 36, 409 Shimizu F., 362, 365 Shimomura S., 314 Shrock R.E., 62, 377, 409, 410 Siggia E.D., 147, 218 Sinai Ya.G., 409 Skierski M., 326 Slavnov N.A., 409 Soni R.P., 101 Sparling G.A.J., 331, 335, 399, 402,

404 Sparrow C., 118 Spiegel E.A., 320 Stegun I.A., 16, 17 Steinberg S., 211, 215, 279, 281 Steudel H., 345, 346, 349, 350, 364 Strampp W., 212, 402 Stroganov Yu.G., 388 Suleimanov B.I., 5, 6, 50

Tabor M., 29, 123, 125, 128, 131, 132, 137, 197, 218, 226, 261, 262, 268, 316-325, 402

Tajiri M., 3, 92, 211 Takano K., 314 Takeno s., 179 Takizawa E.I., 179 Tamizhmani K.M., 211, 212, 214, 215 TangS., 4, 5, 50, 62, 81, 377,

385, 387, 388, 409 Tankersley L.L., 354, 362-366 Templeton S., 393 Thacker H., 62, 409 Thual 0., 320 Thushima K., 211 Tian Chou., 212 Tomei C., 3 Tracy C .A., 13, 50, 62, 377, 380,

382, 387, 407-409, 412, 414-416, 420

Tricomi F.G., 16 Trubowitz E., 3, 249 Tuszynski J.A., 326

Ueno K., 5, 34, 49, 378, 413-415 Umemura H., 314

440

Vaidya H.G., 62, 409 Van de VenA., 298 Van Moerbeke P., 34, 147, 260, 339 Vassiliou P.J., 165, 168, 169 Vessiot E., 165 Vinogradov A.M., 211, 212 Von Gehlen G., 389

Wadati M., 2, 200 Wallach N.R., 250, 251 Wang C.S., 362, 365 Ward R.S., 228, 261, 331, 332, 336,

399 Weinberg E.J., 394, 395, 399 Weinstein A., 253 Weiss J., 125, 128, 129, 131, 132,

137, 141, 160, 197, 202, 218, 226-228, 231, 233-236, 237, 246, 261, 262, 268, 317-320

Wenzel R.G., 364 Wilczek F., 409,415 Wilkinson D., 62, 409 Wils P., 4 Wilson E.B., 17 Wilson G., 249, 253, 416, 418 Winternitz P., 211, 215, 354, 357,

360, 372, 426-429 Witten E., 416 Wolf K.B., 211, 215 Wood D., 218, 319, 402 Woodbury E.J., 362 Wu T.T., 13, 50, 62, 377, 379-382,

387, 407, 409, 410

Yan M.L., 387, 388 Yang C.N., 400 Yang z., 394 Yaremchuk A.I., 429 Yates R.G., 399 Yortsos Y.C., 97 Yoshida H., 149, 151, 153, 155-158

Zajaczkowski W., 163 Zakharov V.E., 3, 333, 335, 348, 399,

428, 429 Zamolodchikov A.B., 377, 387 Zelazny R., 163, 170 Zeng Y.B., 29, 132, 218, 251, 319, 320 Zhiber A.V., 269 Zhou X., 5, 34, 36, 3g 7, 41, 42, 56 Ziglin S.L., 153 Zuber J.B., 407

SUBJECT INDEX

Abel map, 255 affine group, 280, 306 Airy

equation, 57, 83, 85-87, 90, 399 function, 5-7, 10, 49, 58, 86,

89, 94, 308, 313 hierarchy, 91

AKNS equation, 198, 206, 207 system, 334, 335, 403

algebraic geometry, 249 variety,249

AMP symbolic manipulation program, 139

Backlund transformation, 29, 87, 88, 137, 207, 208, 218, 225-227, 230, 232, 233, 235, 237-246, 308, 309, 319, 399, 426

balances principal, 148, 255

Bateman equation, 85 Bessel function, 12, 14, 49,

83,85-87, 89, 90, 101, 179, 308, 313

bi Hamiltonian structure, 212, 215

birational canonical transformation, 303-309

Birkhoff cell, 253-255 decomposition, 253

Boltzman weight, 378, 379, 388 Borel summation method, 63, 71 Boussinesq

equation, 3, 141, 198, 204, 207, 235, 243, 399

modified, 130, 139, 205 Schwarzian, 235

hierarchy modified, 139 Boutroux

asymptotics, 56 transformation, 4, 5, 63, 353,

356 branch point

logarithmic, 149, 316, 317 movable, 62

Burgers equation, 128, 130, 136, 137,

227 complex, 211, 214 conservative, 136 in 2+1 dimension, 211, 215

Burgers (cont'd) Schroedinger type equation, 211,

212, 214, 218, 219

Cart an equivalence problem, 271, 273,

278, 280, 286, 287 invariants, 290-293 matrix, 335 subalgebra, 333, 338

Casimir integral, 237, 243 Cauchy

data, 128 problem, 50, 163, 168, 364

Cauchy-Kovalewsky theorem, 226, 233, 234

caustic surface, 226, 243 Cayley-Hamilton theorem, 256 characteristic manifold, 225, 228 characteristics, 163 Chazy

equation, 130 top, 331,332,339, 340

Chern-Simons electromagnetism, 394 Chern-Simons-Higgs

action, 394 vortex equation, 393-399

Chevalley generators, 335 chiral

fields, 404 Potts models, 378, 387-389

Clarkson equation, 229 classification

of second order ODE, 276 cohomology group, 272 Cole Hopf transformation, 146 compatibility conditions, 264-269 conformal field theory, 377, 386,

387, 407 connection problem, 1, 4, 5, 18,

49, 50, 56, 61-78, 81-96, 381, 382, 384

conservation law, 212, 233, 251 nonlocal, 401

contiguity relations, 303-307 correlation

function, 381, 383, 384, 386, 408, 410

length, 380, 382 critical

point, 104, 105, 128, 187 fixed, 103-122, 128 movable, 97, 128, 380, 382

temperature, 377, 380 curves

elliptic, 294-296

441

Darboux equations, 245 integrability, 166-168, 171, 172 invariant, 167 method, 163-173 transformation

(Backlund), 129, 136, 137, 192-208, 225, 230-235

classical, 137, 198, 200, 201, 203-205, 207, 208, 232

new, 201 Davey Stewartson equation, 399 deformation

equation, 410 theory, 62, 407-413, 415, 419

differential equation, stable, 103-124 invariant, 271, 272

Diophantine equation, 266 Dirac

free equation, 387 field, 408 operator on hyperbolic plane,

408, 416-420 theory in hyperbolic space, 420

direct asymptotic approach, 62 dispersive long wave equation, 425,

427 divisors,

tau, 258-260 Dodd-Bullough-Mikhailov eq., 130 dominances, p, 263 dominant term,

asymptotically, 188 double

asymptotics, 81-90 fibration, 271, 272, 283-294

dressing method, 36 orbit, maximal, 252, 253 transformation, operator, 253-257

Dynkin's diagrams, 94, 309,310, 313

Einstein equation for Bianchi IX comological model, 340

electromagnetism, topologically massive, 394

Ernst equation, 331 Euler-Arnold top (free top), 331,

338 expansion,truncated, 198-200, 319,

320, 324 explicitly linearizable eq., 126

fibration, elliptic, 272, 294 finite

gap integration, 56 point analysis, 262, 265, 267

flag manifold, 251 formal

neighbourhood, 278, 284-287 invariants, 285, 290-293

Frobenius expansion, 72 integrability, 167, 172

Fuchsian differential eq., 62, 412-415 Fuchs' problem, 301 function,

elliptic, 4, 61-68, 74-78, 105, 109, 115, 128, 146, 149, 154, 300

modular, 117 modular, 296

442

function (cont 'd) of wave catastrophes, special, 94 Schwinger, 408,409 sectionally holomorphic, 36 special, 81 tau, 250-260, 307-309, 378, 381,

385, 408-410, 412-420

Gelfand-Levitan-Marchenko integral eq., 5, 13, 17, 56, 176, 371

genus, Riemann surface, 249 Ginzburg- Landau type eq.,

complex, 320, 324-326 global attractors, 373, 375 Gertler vortices, 4 Grassmanian, 253, 417-420 gravitational instanton, 340 gravity, 2D, 36 Green's function, 408, 419-421

Hamiltonian, 250, 252 flow, 226, 233, 235, 238 structure, 225, 233, 244

2nd, 233, 236 systems, 147, 150, 153-159, 242,

246, 299-314 Hamilton-Jacobi eq., 147 Harry Dym equation, 130 heat equation, 128 Heisenberg magnetic spin chain, 190 Henon-Heiles system, 159, 317 hereditary operator, 210 Hermite polynomials, 16, 17, 21,

24, 25, 28 heteroclinic point, 225 Higgs

potential, 394 vortex equation,abelian, 399

Hilbert-Schmidt operators, 9, 14 Hilbert transform, 15 Hirota

bilinear forms, 175-184, 198, 2071 208, 218, 402

method, 175-180, 198 Hirota-Satsuma equation, 139, 198,

207, 208 holonomic

deformations, 301-303, 310 system, 407

homographic differential invariant, 126, 135 group, 125 invariance, 125 invariant, 199, 203 transformation, 126, 134,

197-199, 207 Hopf algebra, 377 hyperbolic space, 407-421 hypergeometric function,

confluent, 308, 313 Gauss, 308, 313, 340

index theory, 396 infinitesimal addition, 259 initial value problem, 345 integer characteristic, 108 integrability, 128, 129, 145, 148,

187, 188, 190, 211, 213 algebraic, 147-149, 151 conditional (partial), 235

integrable equation, 1, 33, 197, 212, 218, 226, 227, 268

integrable system, 113, 230, 237, 345, 346

integral, general, 104, 127 particular, 105 singular, 104, 127

invariant of a differential eq., 271, 272

inverse, monodromic transform (IMT), 34, 58,

97 problem, 40, 98, 101 scattering transform (IST), 1-4, 6,

17, 34, 38, 97, 145-147, 217' 345-347' 364, 370-372, 375

with variable spectral parameter, 429

involutive system in Cartan sense, 169 Ising

field theory, massive, 407 model, 4, 377-389, 409, 410, 417 quantum spin chain, 378, 388 spin chain, transverse, 383-386

isomonodromic deformation, 5, 6, 13, 34, 36, 57,

62, 81-96, 301, 407' 408, 413 problem, 58, 407-421 condition, 55, 58, 90, 94, 97 deformation method (IDM), 49-58, 81,

83, 86 Ito eq. (in 2+1 dimensions), 208

Jacobian, 249 Jacobi inversion theorem, 259 Jest function, 201, 348, 349

Kac-Moody algebra, 175 Kac-Moody-Virasoro algebra, 427, 428 Kac-Von Moerbeke

equation, 34, 35 system, 238, 239

Kadomtsev-Petviashvili eq., 337, 399, 402, 404

Klein-Gordon equation, 167, 410 field, 408

Korteweg-de Vries (KdV) equation, 2, 33-35, 90-92, 130, 131,

176, 178, 193, 198, 200, 202, 225, 226, 228, 230, 232-234, 237, 239, 243, 331, 334, 335, 399-404, 410

higher, 90, 91, 235 modified (mKdV), 1, 2, 4-6, 146,

231-234, 386, 399 Schwarzian, 228, 232, 237

hierarchy, 91, 198, 319, 335 Kowalewskaia

test, 188 top (heavy top), 331, 339

Kowalevski exponents, 147, 150-152, 157, 158, 176, 180

Kruskal Ansatz (gauge), 134, 136, 158-160, 262

Kuramoto-Sivashinsky equation, 130

Laplace-Darboux transform, 226, 243-246

Laplace transform, 49, 56, 58, 63, 71-73, 86, 88

hyperbolic, 408-415

Laurent series, 105, 125, 128-131, 135, 136, 147, 149, 188, 191, 197, 218-221, 230, 233-235, 251, 309, 315, 316, 318

Lax pair, 35, 95, 128, 129, 137, 138, 197-200, 202-207, 218, 225-227, 230, 232, 235, 319, 332, 334, 335, 339, 354, 364, 370, 378, 379, 399, 401, 403, 429

leading order analysis, 262-264 Legendre

polynomials, 373 transformation, 236

Lenard recursion formula, 234 Lie

algebra, 332-334, 337, 355, 379 A( ) , 337 su(l,l), 411 su (n), 333, 334, 338

bracket, 332 derivative, 170 group,

infinite dimensional, 212, 249 point, 33, 355

subgroup of GL(3,C), 271, 273-278 symmetries, 315, 319

point, 211-213, 215 theory, 249

limiting procedures, 35 linear problem, 49 Liouville

equation, 129, 168, 269, 331 integrability, 152

Loop group, 254 Lorenz equation, 118-122, 318 Latka - Volterra system, 151, 152

MACSYMA, symbol manipulation program, 215, 372

map, submersive, 258, 426-428 material excitation, 346, 354, 367 matrix,

circulant, 237, 243, 244 factorization, 250

Maxwell - Bloch system, pumped, 425, 429, 430

Miura transformation, 137, 138, 141, 205, 225, 230, 232, 233, 235

Mobius group, 126, 134, 197, 225, 230, 231,

233-235 transformation, 97, 126, 184, 342

Monge characteristic, 165, 168, 172 monodromy

data, 34, 37, 40, 42, 50, 51, 56, 57, 84-86, 90, 98, 100

formal, 92 preserving deformation (see

isomonodromic deformations) monopoles, 412 multiple scale

analysis, 63, 65-68 expansion, 56, 190, 357

Nahm equation, 331, 332, 339 natural boundaries, movable, 128, 332 Newell- Whitehead eq., real, 319,

321-324, 326-329 Nicholson Watson Langer formulae, 86 nonlinear

ladder equation, 336

443

nonlinear (cont'd) ladder equation (cont'd)

continuous version, 337 optics, 4 Schroedinger equation (NLS), 3, 179,

189, 331, 334, 335, 375, 399, 404, 425, 426

derivative, 1, 3, 4, 17 splitting (coupled), 92

Schrodinger hierarchy, splitting (coupled), 93, 335

oscillator, damped, 315 special function, 49

n - point correlations, 378 function, 380, 407, 408

n - wave equation, 331, 334

orthogonal polynomials, 36, 44, 45 order parameter, 379

Painleve, alpha method, 303 analysis, 125-143, 152, 164, 173,

191, 198, 202, 204, 211, 217-221, 249, 315, 396-399, 430

invariant, 125-143, 197-200 balance, 249-260 I eq., 2-5, 33, 35, 36, 44, 61-78,

82, 90, 91, 97, 112, 128, 146, 176, 179, 192, 301, 308, 313, 409

asymptotics, 61-78 discrete, 44 hierarchy, 91, 93

II eq., 1-12, 20, 27, 34, 36, 37, 40, 42, 43, 50, 51, 55, 57, 82, 83, 85, 86, 88, 92, 97, 98, 110, 128, 146, 147' 176, 180-184, 281-283, 303-310, 313, 385, 386, 398, 399, 410

asymptotics, 55 direct problem, 37, 42 hierarchy, 93, 94 inverse problem, 42

III eq., 1-5, 12-15, 20, 27, 82-84, 87' 89, 97-101, 114, 128, 146, 149, 175-177, 180-184, 301, 303, 307' 313, 345, 347, 350, 373, 380-382, 386, 393, 396, 409

inverse problem, 98, 100, 101 direct problem, 98-100

IV eq., 1-5, 16-29, 35, 36, 82, 97, 116, 128, 146, 173, 180-184, 303, 305, 307, 313, 409, 426, 428, 429

v eq., 2, 82, 97, 98, 114, 128, 146, 176, 180-184, 303, 307, 310, 313, 353, 354, 356, 359, 373, 381, 384-386, 409, 428, 429

asymptotics, 353, 354, 356-359 perturbed, 359

VI eq., 2, 82, 97, 98, 128, 146, 182, 184, 272, 302, 303, 307' 313, 381, 428, 409, 410, 415

and elliptic fibrations, 297 11 eq., 279 17 eq., 279

444

Painleve (continued) i2 eq., 280, 281 34 eq., 82, 92, 281

hierarchy, 93, 94 37 eq., 279 41 eq., 279 43 eq., 279 classification, 175, 279, 294

of PDEs, 261-269 conjecture, 2, 145, 148, 218 Darboux equation, 132, 136,

138-149, 198, 200-208 equations, 1, 2, 4, 5, 18, 29,

33-43, 49-58, 62, 85, 94, 146, 175-184, 226, 271, 280-284, 296, 299-314, 332, 395, 398, 409, 425-430

asymptotic properties, 49-56, 81

admissible, 261-269 discrete, 33, 36, 44-46 fixed singular points, 176, 177 Hamiltonians, 94 Hirota form, 175-184 limits, 35, 82, 97 perturbed, 395, 396 symmetries of, 279

integrability, 145-160 limits, 82 method, poly, 158-160 property, 2, 4, 97, 125-143, 145,

147-149, 152, 153, 160, 186, 187, 211-221, 225-246, 261, 271, 272, 275, 294, 317, 318, 396, 397, 399

conditional, 320 weak, 148, 150, 152, 153, 160

rank, 302 test, 125-129, 132, 143, 168,

187-194, 197, 199, 261, 262, 267, 268, 426

transcendents, 49, 50, 70, 71, 78, 81-96, 115, 127' 128, 226, 272, 297, 347, 361, 373, 375, 377-389, 420

asymptotics, 81-96 parabolic cylinder function

(Weber-Hermite), 16-18, 20, 21, 23, 28, 29, 53, 54, 91, 94, 308, 311

partition function, 378 Pauli matrices, 37, 334, 383, 389 periodic

boundary condition, 249 fixed point, 225, 226, 233, 237-246

perturbation theory, singular, 190 Pfaffian system, 169, 170, 273 Pinney equation, 139-143 Plucker coordinates, 250, 255, 256 Poincare disk, 408, 410-412 polyelectrolytes, 4 power series, generalized, 188, 189,

191, 192 projective

connection, 272, 284, 294, 295 space, 252-255

prolongation, 212, 273 pseudopotential, Riccati type, 198,

202 Psi series, 225, 227, 229, 316-318,

320 Puiseux series, 152, 153 pump wave, 345, 347, 349, 354, 362,

367, 369, 374

quantum deformation parameter, 379 field theory, 49

holonomic, 407, 408 in hyperbolic space, 408, 412 20, 407-417 bosonic, 410, 414, 415 fermionic, 415, 416 massive, 407

gravity, 410 group,377, 379, 387, 388 spin Hamiltonian, 379

Raman effect, 353, 361 scattering spontaneous, 361

Rand equation, 230 recursion operator, 212, 213, 225,

401, 404 REDUCE, symbol manipulation

program, 139, 215 rescaling

Ansatz, 316, 317, 325, 326 transformation, 316-318, 321-324

reduction , 393, 399, 400-402 resonance, 129, 130, 132, 138-140,

147, 148, 152, 153, 157, 176, 189-191, 199, 206, 218-221, 229, 230, 264, 315, 317-320, 397

analysis, 130, 138, 264-267 negative, 190

nonintegral, 190, 191 Riccati

equation, 29, 126, 128, 137, 143, 146, 198, 202-204, 217, 232, 269, 307

system, 198, 199, 202 Riemann

double wave, 163-174 function, 163, 168, 169 Hilbert problem, 36-43, 50,

56-58, 88, 100, 101 invariant, 163, 169

generalized, 163 surface, 154, 156, 157, 249, 265 theta function, 249, 255 wave, 163-173

Rikitake eq., 148, 149

scaling limit, 380, 381 symmetry, 317, 353

scattering data, time evolving, 370 problem, 200, 201

Schlesinger equation, 410, 413 transform, 29, 37, 87

Schrodinger equation, 200-202, 335, 403

Schubert cycle, 260 Schwarz ian

equation, 126, 137, 142, 143, 234, 235

derivative, 228 derivative, 225, 228, 231, 232, 235,

319 triangle function, 117, 341

Schwarz reflection, 37 Schwinger function, 408, 409 self-similar solution (similarity

solution), 7, 49, 355, 372-375

sigma model, nonlinear, 331 similarity reduction, 2, 4, 17, 19,

33, 92-94, 146, 211, 216-218, 317-321, 323, 325-327, 429

of discrete equations, 34, 35 Sine Gordon equation, 1, 3, 4, 13,

189, 269, 331, 345-247, 399 in N-dimensions, 236

singularity analysis, 148, 175 apparent, 302 exponential movable, 190-193 manifold, 218, 225-246 structure, 145, 316

singular manifold, 125, 131, 132, 134, 137,

141, 197, 198, 225-246, 262, 263, 318-321

movable, 128, 131, 135 point

analysis, 103, 104, 262-264, 267 fixed (intrinsic), 61, 67, 72, 73,

104, 127, 226 irregular, 5, 38, 192 movable (parametric), 50, 104,

128, 129, 187, 229 regular, 5, 38, 302

slowly varying envelope approximation, 362, 363

soliton solution, 1, 176, 178, 179, 202,

213, 346, 349, 354, 364-366, 369, 397, 429

system, 211, 217 theory, 249, 253

solution, general single valued, 187

spin system, 377-389 spontaneous magnetization, 382 statistical mechanic, 61, 62,

377-389 solvable model, 377, 379, 386,

387 Stimulated Raman Scattering (SRS)

equation, 345-350, 353-375 asymptotic solution, 345-350,

357-360 similarity solution, 345-347, 355 transient limit, 353, 354

Stokes lines, 52 matrices, 50, 54, 57, 101 multipliers, 39, 50, 83 nonlinear phenomena, 56 rays, 81 wave, 345-347, 353, 354, 362-365,

369, 374 string

equation, massive (2n+l,2), 91 theory, 61, 62

structures, G, 273, 275 e, 271, 275-277, 279, 282

subalgebras, 426-428 subgroup, 426-428

parabolic, 253 superconductivity, high T, 394 symmetric space, 408 symmetries, 211-221

generalized (Lie- Backlund), 212, 213

potential, 212 symmetry

breaking, 317 condition, 57

445

symmetry ( cont 'd) discrete, 230, 232, 235 infinitesimal generator, 212 master, 212-214 nonlocal, 212 reduction, 331-342, 353, 372, 425

classical, 321, 324, 325 non classical, 321, 324,

326-329

Taylor series, 52, 131, 242 theory of covering, 212 time dependent eigenvalues, 346,

347, 350 Toda

chain, 399 constants of motion, 251 equation, 309, 383 lattice, 226, 243, 245, 246, 331,

336 complex SL(l+l,C), 249-260

molecule, 331, 335, 336 continuous version, 337

Toeplitz determinant, 383 trace class operator, 416 transcendental

degree, 307 element, 300

transfer matrix, 378, 379, 387, 388 transformation,

contact, 425, 429, 430 fiber preserving, 271-273, 275,

276, 279, 281, 283 gauge, 383, 399, 400 infinitesimal, 355, 358 point, 271, 355, 358, 428, 430 similarity, 150, 334

446

trilinear form, 402 turning point, 52-54, 81, 88, 89 twister space, 272-278

vector field, 211 8-vertex model, 379 vertical connection, 81 Vessiot distribution, 164 Virasoro algebra, 427 vortex solutions, 305, 396, 399

Ward conjecture, 331 3-wave equation, 333, 425, 428, 429 Weyl

chamber, 307 group, 253

affine, 306, 313 Whitham's method, 56 WKB method, 51, 52, 54-56, 90

X-Y model, 409,410

Yang Baxter equation, 379 Yang Mills equation,

selfdual, 331-342, 393, 399-404

Yoshida theorem, 149-153, 156, 157

Zakharov-Shabat eigenvalue problem, 348

zero curvature condition, 400 Ziglin theory of nonintegrability,

153-158