Geodesic
Asymptotics at $t\to\infty$ of the solution to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as $x\to\pm\infty$
Teoretičeskaâ i matematičeskaâ fizika, Tome 78 (1989) no. 3, pp. 345-356 
R. F. Bikbaev; R. A. Sharipov. Asymptotics at $t\to\infty$ of the solution to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as $x\to\pm\infty$. Teoretičeskaâ i matematičeskaâ fizika, Tome 78 (1989) no. 3, pp. 345-356. http://geodesic.mathdoc.fr/item/TMF_1989_78_3_a3/
@article{TMF_1989_78_3_a3,
     author = {R. F. Bikbaev and R. A. Sharipov},
     title = {Asymptotics at~$t\to\infty$ of~the solution to~the {Cauchy} problem for the {Korteweg{\textendash}de~Vries} equation in~the class of~potentials with finite-gap behavior as~$x\to\pm\infty$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {345--356},
     year = {1989},
     volume = {78},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1989_78_3_a3/}
}
TY  - JOUR
AU  - R. F. Bikbaev
AU  - R. A. Sharipov
TI  - Asymptotics at $t\to\infty$ of the solution to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as $x\to\pm\infty$
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1989
SP  - 345
EP  - 356
VL  - 78
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1989_78_3_a3/
LA  - ru
ID  - TMF_1989_78_3_a3
ER  - 
%0 Journal Article
%A R. F. Bikbaev
%A R. A. Sharipov
%T Asymptotics at $t\to\infty$ of the solution to the Cauchy problem for the Korteweg–de Vries equation in the class of potentials with finite-gap behavior as $x\to\pm\infty$
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1989
%P 345-356
%V 78
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1989_78_3_a3/
%G ru
%F TMF_1989_78_3_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

The Cauchy problem for the KdV equation is considered in the class of functions approaching at $x\to\pm\infty$ two different finite-gap solutions of this equation which correspond to the same Riemann surface. With the help of the inverse scattering method the large time asymptotics of the Cauchy problem solution is considered.

[1] Zakharov V. E., Manakov S. V., ZhETF, 71:1 (1976), 203–215 | MR

[2] Shabat A. B., DAN SSSR, 211:6 (1973), 1310–1313 | MR | Zbl

[3] Ablowitz M., Segur H., Studies in Appl. Math., 57, 1977, 13–44 | DOI | MR | Zbl

[4] Novokshenov V. Yu., DAN SSSR, 251:4 (1980), 799–802 | MR | Zbl

[5] Buslaev V. S., Sukhanov V. V., Zap. nauch. seminarov LOMI, 120, 1982, 32–51 | MR

[6] Its A. R., DAN SSSR, 261:1 (1981), 14–18 | MR | Zbl

[7] Kuznetsov E. A., Mikhailov A. V., ZhETF, 67:11 (1974), 1717–1727

[8] Khruslov E. Ya., Mat. sbornik, 99(141):2 (1976), 261–281 | MR | Zbl

[9] Kotlyarov V. P., Khruslov E. Ya., TMF, 68:2 (1986), 172–186 | MR

[10] Its A. R., Ustinov A. F., DAN SSSR, 291:1 (1986), 91–95 | MR | Zbl

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

[12] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[13] Dubrovin B. A., UMN, 36:2 (1981), 11–80 | MR | Zbl

[14] Rodin Yu. L., Physica D, 24D (1987), 1–53 | DOI | MR | Zbl

[15] Krichever I. M., Funkts. analiz i ego prilozh., 9:2 (1975), 77–78 | MR | Zbl

[16] Sharipov R. A., UMN, 41:5 (1986), 203–204 | MR | Zbl

[17] Fay J., Lecture Notes in Math., 352, Springer-Verlag, 1973 | DOI | MR | Zbl

[18] Kaup D., SIAM J. of Appl. Math., 31 (1976), 121–129 | DOI | MR

[19] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, Nauka, M., 1974 | MR