``Isomonodromy'' solutions of equations of zero curvature

A. R. Its

Izvestiya. Mathematics , Tome 26 (1986) no. 3, pp. 497-529.

Résumé

A detailed discussion is given of the analytic properties (explicit description, asymptotic representations, etc.) of a new class of solutions of completely integrable evolution systems – the class of “isomonodromy solutions” – introduced in recent works of a group of Japanese mathematicians: M. Sato, M. Jimbo, T. Miwa, and others. The role of the concept of an isomonodromy solution is demonstrated in such important questions of the theory of equations of zero curvature as finding the time asymptotics of solutions of corresponding Cauchy problems in the class of rapidly decreasing initial data. Bibliography: 30 titles.

Bibliographie

[1] Jimbo M., Miwa T., Sato M., Holonomic Quantum Fields, preprint No 305, Res. Inst. Math. Sci., Kyoto, 1979

[2] Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I, preprint No 319, Res. Inst Math. Sci., Kyoto, 1980 | MR | Zbl

[3] Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, preprint No 327, Res. Inst. Math. Sci., Kyoto, 1980 | Zbl

[4] Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations, preprint No 394, Res. Inst. Math. Sci., Kyoto, 1982 | MR

[5] Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskii L. P., Teoriya solitonov, Nauka, M., 1980 | MR

[6] Krichever I. M., “Avtomodelnye resheniya uravnenii tipa KdF”, Funkts. analiz i ego prilozh., 14:3 (1980), 83–84 | MR | Zbl

[7] Flashka H., Newell A. C., “Monodromy and spectrum preserving deformation”, Commun. Math. Phys., 76 (1980), 65–87 | DOI | MR

[8] Its A. R., “Asimptotika reshenii nelineinogo uravneniya Shredingera i izomonodromnye deformatsii sistem lineinykh differentsialnykh uravnenii”, Dokl. AN SSSR, 261:1 (1981), 14–18 | MR | Zbl

[9] Its A. R., Petrov V. E., “Izomonodromnye resheniya uravneniya Sine-Gordon i vremennaya asimptotika ego bystroubyvayuschikh reshenii”, Dokl. AN SSSR, 265:6 (1982), 1302–1304 | MR

[10] Zakharov V. E., Manakov S. V., “Asimptoticheskoe povedenie nelineinykh volnovykh sistem, integriruemykh metodom obratnoi zadachi”, ZhETF, 71:1 (1976), 203–215 | MR

[11] Ablowitz M. J., Segur H., “Asymptotic solutions of the Korteweg-de Vries equation”, Stud. Appl. Math., 57 (1977), 13–44 | MR | Zbl

[12] Miles J. W., “The asymptotic solution of the Korteweg-de Vries equation”, Stud. Appl. Math., 60:1 (1979), 59–72 | MR

[13] Buslaev V. S., Sukhanov V. V., “Ob asimptoticheskom povedenii pri reshenii uravneniya $\psi_{xx}+u\psi+\dfrac\lambda4\psi=0$ s potentsialom $u$, udovletvoryayuschim uravneniyu KdF”, Problemy matematicheskoi fiziki, 10, LGU, 1982

[14] Novokshenov V. Yu., “Asimptotika pri $t\to\infty$ reshenii zadachi Koshi dlya nelineinogo uravneniya Shredingera”, Dokl. AN SSSR, 251:4 (1980), 779–802 | MR

[15] Novokshenov V. Yu., “Asimptoticheskie formuly dlya reshenii sistemy nelineinykh uravnenii Shredingera”, Uspekhi matem. nauk, 37:2 (1982), 215–216 | MR | Zbl

[16] Shabat A. B., “Obratnaya zadacha rasseyaniya dlya sistemy differentsialnykh uravnenii”, Funkts. analiz i ego prilozh., 9:3 (1975), 75–78 | Zbl

[17] Zakharov V. E., Shabat A. B., “Integrirovanie nelineinykh uravnenii matematicheskoi fiziki metodom obratnoi zadachi rasseyaniya”, Funkts. analiz i ego prilozh., 13:3 (1979), 13–22 | MR | Zbl

[18] Zakharov V. E., Mikhailov A. V., “Relyativistski-invariantnye dvumernye modeli teorii polya, integriruemye metodom obratnoi zadachi”, ZhETF, 74:6 (1978), 1953–1973 | MR

[19] Mikhailov A. V., “O gruppe reduktsii, dvumerizovannoi tsepochke Toda i zadache Rimana”, Trudy seminara ITEF po teorii yadra, 1979 ; препринт ИТЭФ–41, 1980 | Zbl

[20] Krichever I. M., “Analogi formuly Dalambera dlya uravnenii glavnogo kiralnogo polya i uravneniya Sine-Gordon”, Dokl. AN SSSR, 253:2 (1980), 288–292 | MR | Zbl

[21] Its A. R., Tochnoe integrirovanie v rimanovykh $\theta$-funktsiyakh nelineinogo uravneniya Shredingera i modifitsirovannogo uravneniya KdF, avtoreferat kandidatskoi dissertatsii, 1977

[22] Its A. R., Kotlyarov V. P., “Ob odnom klasse reshenii nelineinogo uravneniya Shredingera”, Dokl. AN USSR, seriya A, 1976, no. 11, 965–968 | MR | Zbl

[23] Cherednik I. V., “Ob usloviyakh veschestvennosti v “konechnozonnom” integrirovanii”, Dokl. AN SSSR, 252:5 (1980), 1104–1107 | MR

[24] Takhtadzhyan L. A., “Gamiltonovy sistemy, svyazannye s uravneniem Diraka”, Zapiski nauchnykh seminarov LOMI, 37, 1973, 66–76 | Zbl

[25] Manakov S. V., “Nelineinaya difraktsiya Fraungofera”, ZhETF, 65 (1973), 1392–1403

[26] Zakharov V. E., Manakov S. V., “O rezonansnom vzaimodeistvii volnovykh paketov v nelineinykh sredakh”, Pisma v ZhETF, 18:7 (1973), 413–447

[27] Beitman G., Erdeii A., Vysshie transtsendentnye funktsii, II, Nauka, M., 1974

[28] Ablowitz M. J., “Lectures on the inverse scattering transform”, Stud. Appl. Math., 58 (1978), 17–94 | MR | Zbl

[29] Vazov V., Asimptoticheskie razlozheniya reshenii obyknovennykh differentsialnykh uravnenii, Mir, M., 1968

[30] Uizem Dzh., Lineinye i nelineinye volny, Mir, M., 1977