Geodesic
Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a two-dimensional generalization of the Toda lattice
Izvestiya. Mathematics, Tome 24 (1985) no. 2, pp. 347-382 
V. Yu. Novokshenov. Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a two-dimensional generalization of the Toda lattice. Izvestiya. Mathematics, Tome 24 (1985) no. 2, pp. 347-382. http://geodesic.mathdoc.fr/item/IM2_1985_24_2_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The leading term of the asymptotics of a solution of the nonlinear hyperbolic system $$ \square u_n=\exp(u_{n+1}-u_n)-\exp(u_n-u_{n-1}),\qquad n=1,2,\dots,N, $$ for large times is constructed and justified. A version of the method of the inverse problem reducing to the solution of a matrix problem of linear conjugation on the complex plane of the spectral parameter is used to solve this system. The coefficients of the asymptotics of $u_n$ are expressed explicitly in terms of the elements of the Riemann matrix realizing the linear conjugation. A theorem is proved on the approximation of the exact solution by the asymptotics constructed. Bibliography: 14 titles.

[1] S. P. Novikov (red.), Teoriya solitonov, Nauka, M., 1980 | MR

[2] Zakharov V. E., Manakov S. V., “Asimptoticheskoe povedenie nelineinykh volnovykh sistem, integriruemykh metodom obratnoi zadachi”, ZhETF, 71:1(7) (1976), 203–215 | MR

[3] Novokshenov V. Yu., “Asimptotika pri $t\to\infty$ resheniya zadachi Koshi dlya nelineinogo uravneniya Shredingera”, Dokl. AN SSSR, 251:4 (1980), 799–801 | MR

[4] Novokshenov V. Yu., Khabibullin I. T., “Nelineinye differentsialno-raznostnye skhemy, integriruemye metodom obratnoi zadachi rasseyaniya. Asimptotika resheniya pri $t\to\infty$”, Dokl. AN SSSR, 257:3 (1981), 543–547 | MR | Zbl

[5] Buslaev V. S., “Ispolzovanie determinantnogo predstavleniya reshenii uravneniya Kortevega–de Friza dlya issledovaniya ikh asimptoticheskogo povedeniya pri bolshikh vremenakh”, UMN, 36:4 (1981), 217–218

[6] 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

[7] Zhiber A. V., Shabat A. B., “Uravneniya Kleina–Gordona s netrivialnoi gruppoi”, Dokl. AN SSSR, 247:5 (1979), 1103–1107 | MR

[8] Mikhailov A. V., “The reduction problem and the inverse scattering method”, Physica 3D, 1,2 (1981), 73–117

[9] Mikhailov A. V., “Ob integriruemosti dvumernogo obobscheniya tsepochki Tody”, Pisma v ZhETF, 30:7 (1980), 443–447

[10] Shabat A. B., “Obratnaya zadacha rasseyaniya”, Differents. uravneniya, 15 (1979), 1824–1835 | MR

[11] Beitmen G., Erdeii L., Vysshie transtsendentnye funktsii, t. 2, Nauka, M., 1973, 296 pp.

[12] Fedoryuk M. V., Metod perevala, Nauka, M., 1977, 368 pp. | MR

[13] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977, 640 pp. | MR

[14] Ablowitz M., Ramani A., Segur H., “A connection between nonlinear evolution equations and ordinary differential aquations of $P$-type, II”, J. Math. Phys., 21:5 (1980), 1006–1015 | DOI | MR | Zbl