Some new nonlinear evolution equations integrable by the inverse problem method

V. K. Mel'nikov

Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 461-489

Résumé

Several new nonlinear evolution equations integrable by the inverse problem method are obtained. The method applied in finding these equations is believed to be essentially new. The comparison of that method with other methods for finding nonlinear evolution equations integrable by the inverse problem method is given. In particular, it is shown that the methods using the Heisenberg equation (the so-called Lax representation) are not suitable to obtain the equations studied here. Bibliography: 23 titles.

Bibliographie

[1] Gardner C. S., Green J. M., Kruskal M. D., Miura R. M., “Method for solving the KdV equation”, Phys. Rev. Lett., 19:19 (1967), 1095–1097 | DOI | Zbl

[2] Melnikov V. K., “Ob uravneniyakh, porozhdaemykh operatornym sootnosheniem”, Matem. sb., 108 (150) (1979), 378–392 | MR

[3] Melnikov V. K., “Metod obratnoi zadachi v teorii nelineinykh evolyutsionnykh uravnenii”, EChAYa, 11:5 (1980), 1224–1272 | MR

[4] Melnikov V. K., “Zakony sokhraneniya dlya odnogo klassa sistem nelineinykh evolyutsionnykh uravnenii”, Funkts. analiz, 15:1 (1981), 43–60 | MR

[5] Gardner C. S., Green J. M., Kruskal M. D., Miura R. M., “Korteweg-de Vries equation? and generalization. VI. Method for exact solution”, Comm. Pure Appl. Math., XXVII:1 (1974), 97–133 | DOI | MR

[6] Marchenko V. A., “Periodicheskaya zadacha Kortevega–de Friza”, Matem. sb., 95 (137), 331–356 | Zbl

[7] Lax P., “Periodic solutions of the KdV equations”, Lectures in Appl. Math., 15 (1974), 85–96 | MR

[8] Gelfand I. M., Dikii L. A., “Asimptotika rezolventy shturm-liuvillevskikh uravnenii i algebra uravnenii Kortevega–de Friza”, UMN, XXX:5 (1975), 67–100 | MR

[9] Gelfand I. M., Dikii L. A., “Drobnye stepeni operatorov i gamiltonovy sistemy”, Funkts. analiz, 10:4 (1976), 13–39 | MR

[10] Gelfand I. M., Dikii L. A., “Rezolventa i gamiltonovy sistemy”, Funkts. analiz, 11:2 (1977), 11–27 | MR

[11] Gelfand I. M., Dikii L. A., “Ischislenie strui i nelineinye gamiltonovy sistemy”, Funkts. analiz, 12:2 (1978), 8–23 | MR

[12] Ablowitz M. J., Kaup D. J., Newell A. S., Segur H., “Nonlinear-evolution equations of physical significance”, Phys. Rev. lett., 31:2 (1973), 125–127 | DOI | MR

[13] Lax P., “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math., 21:5 (1968), 467–490 | DOI | MR | Zbl

[14] Zakharov V. E., “The inverse scattering method”, Solutions, eds. Bullough R. K., Caudrey P. J., Springer-Verlag, Berlin–Heidelberg–New York, 1980, 243–285

[15] Makhankov V. G., “On stationary solutions of the Schrödinger equation with a selfconsistent potential satisfying Boussinesg's equation”, Phys. Lett., 50A:1 (1974), 42–44 | MR

[16] Makhankov V. G., “Solitony i chislennyi eksperiment”, EChAYa, 14:1 (1983), 123–180 | MR

[17] Wilson G., “The affine Lie algebra $C_2^{(1)}$ and an equation of Hirota and Satsuma”, Phys. Lett., 89A:7 (1982), 332–334 | MR

[18] Melnikov V. K., “O nekotorykh novykh nelineinykh evolyutsionnykh uravneniyakh, reshaemykh metodom obratnoi zadachi”, UMN, 37:4 (1982), 111

[19] Gombash P., Problema mnogikh chastits v kvantovoi mekhanike, IL, M., 1952

[20] Miura R. M., “Korteweg–de Vries equation and generalizations. I. A remarkable explicit transformation”, J. Math. Phys., 9:8 (1968), 1202–1204 | DOI | MR | Zbl

[21] Melnikov V. K., O preobrazovaniyakh nelineinykh evolyutsionnykh uravnenii, analogichnykh preobrazovaniyu Miury, Preprint OIYaI R2-81-205, OIYaI, Dubna, 1981 | MR

[22] Melnikov V. K., “Nekotorye novye vpolne integriruemye modeli samosoglasovannogo polya”, Problemy fiziki vysokikh energii i kvantovoi teorii polya, T. 1 (Trudy V Mezhdunarodnogo seminara), Protvino, 1982, 93–114

[23] Melnikov V. K., Ob uravneniyakh, opisyvayuschikh vzaimodeistvie voln, Preprint OIYaI R2-82-684, OIYaI, Dubna, 1982