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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

[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