Geodesic
Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type
Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 229-248 
E. Ya. Khruslov. Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type. Sbornik. Mathematics, Tome 28 (1976) no. 2, pp. 229-248. http://geodesic.mathdoc.fr/item/SM_1976_28_2_a6/
@article{SM_1976_28_2_a6,
     author = {E. Ya. Khruslov},
     title = {Asymptotics of the solution of the {Cauchy} problem for the {Korteweg{\textendash}de~Vries} equation with initial data of step type},
     journal = {Sbornik. Mathematics},
     pages = {229--248},
     year = {1976},
     volume = {28},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_28_2_a6/}
}
TY  - JOUR
AU  - E. Ya. Khruslov
TI  - Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type
JO  - Sbornik. Mathematics
PY  - 1976
SP  - 229
EP  - 248
VL  - 28
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1976_28_2_a6/
LA  - en
ID  - SM_1976_28_2_a6
ER  - 
%0 Journal Article
%A E. Ya. Khruslov
%T Asymptotics of the solution of the Cauchy problem for the Korteweg–de Vries equation with initial data of step type
%J Sbornik. Mathematics
%D 1976
%P 229-248
%V 28
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1976_28_2_a6/
%G en
%F SM_1976_28_2_a6

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

The method of the inverse scattering problem is used to solve the Cauchy problem for the Korteweg–deVries equation with initial data of step type: $u(x,0)\to-c^2$ ($x\to-\infty$), $u(x,0)\to0$ ($x\to\infty$). Formulas are obtained for transforming the scattering data with respect to the time, making it possible to obtain a solution $u(x,t)$ of the problem for arbitrary $t$ with the aid of linear integral equations of scattering theory. The asymptotic behavior of the solution as $t\to+\infty$ is investigated in a neighborhood of the wave front $\bigl(x>4c^2t-\frac1{2c}\ln t^N\bigr)$. It is shown that in this region the solution splits up into solitons, the distance between which increases as $\ln t^{1/c}$, and an explicit form for these solitons is derived. Bibliography: 12 titles.

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

[2] P. D. Laks, “Integraly nelineinykh evolyutsionnykh uravnenii i uedinennye volny”, Matematika, 13:5 (1969), 128–150

[3] V. A. Marchenko, “Periodicheskaya zadacha Kortevega-de Friza”, Matem. sb., 95 (137) (1974), 331–356 | Zbl

[4] S. P. Novikov, “Periodicheskaya zadacha dlya uravneniya Kortevega-de Friza”, Funkts. analiz, 8:3 (1974), 54–66 | MR | Zbl

[5] A. R. Its, V. B. Matveev, “Operatory Shredingera s konechnozonnym spektrom i $N$-solitonnye resheniya uravneniya Kortevega-de Friza”, Teor. matem. fizika, 23:1 (1975), 51–68 | MR

[6] V. E. Zakharov, “Metod obratnoi zadachi rasseyaniya”, glava v knige: I. A. Kunin, Teoriya uprugosti sred s mikrostrukturoi, izd-vo «Nauka», Moskva, 1974

[7] E. Ya. Khruslov, “Raspad nachalnogo vozmuscheniya tipa stupenki v uravnenii Kortevega-de-Friza”, Pisma v ZhETF, 21:8 (1975), 469–472

[8] A. V. Gurevich, L. P. Pitaevskii, “Raspad nachalnogo razryva v uravnenii Kortevega-de Friza”, Pisma v ZhETF, 17:5 (1973), 268–271

[9] A. V. Gurevich, L. P. Pitaevskii, “Nestatsionarnaya struktura besstolknovitelnoi udarnoi volny”, ZhETF, 65:2(8) (1973), 590–604 | MR

[10] A. B. Shabat, “Ob uravneniyakh Kortevega-de Friza”, DAN SSSR, 211:6 (1973), 1310–1313 | Zbl

[11] M. J. Ablowitz, A. C. Newell, “The decay of the continuous spectrum for solutions of the Korteweg-de Vries equation”, J. Math. Phys., 149 (1973), 1277–1284 | DOI | MR

[12] V. Buslaev, V. Fomin, “K obratnoi zadache rasseyaniya dlya odnomernogo uravneniya Shredingera na vsei osi”, Vestnik LGU, 1:1 (1962), 56–64 | MR | Zbl