Geodesic
Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the “threshold type”
Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 2, pp. 214-227 Cet article a éte moissonné depuis la source Math-Net.Ru

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We derive a new integral equation that linearizes the Cauchy problem for the Korteweg–de Vries equation for the initial condition of the threshold type, where the initial function vanishes as $x\to-\infty$ and tends to some periodic function as $x\to+\infty$. We also expand the solution of the Cauchy problem into a radiation component determined by the reflection coefficient and a component determined by the nonvanishing initial condition. For the second component, we derive an approximate determinant formula that is valid for any $t\ge 0$ and $x\in(-\infty,X_N)$, where $X_N\to\infty$ with the unboundedly increasing parameter $N$ that determines the finite-dimensional approximation to the integral equation. We prove that as $t\to\infty$, the solution of the Cauchy problem in the neighborhood of the trailing edge decays into asymptotic solitons, whose phases can be explicitly evaluated in terms of the reflection coefficient and other parameters of the problem. 
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     author = {V. B. Baranetskii and V. P. Kotlyarov},
     title = {Asymptotic behavior in the trailing edge domain of the solution of the {KdV} equation with an initial condition of the {\textquotedblleft}threshold type{\textquotedblright}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     volume = {126},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_126_2_a3/}
}
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V. B. Baranetskii; V. P. Kotlyarov. Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the “threshold type”. Teoretičeskaâ i matematičeskaâ fizika, Tome 126 (2001) no. 2, pp. 214-227. http://geodesic.mathdoc.fr/item/TMF_2001_126_2_a3/

[1] C. S. Gardner, J. M. Green, M. D. Kruskal, R. M. Miura, Phys. Rev. Lett., 19 (1967), 1095 | DOI | Zbl

[2] V. E. Zakharov, A. B. Shabat, ZhETF, 61 (1971), 118

[3] A. B. Shabat, DAN SSSR, 211 (1973), 1310 | Zbl

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

[5] M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981 | MR | Zbl

[6] V. E. Zakharov, S. V. Manakov, ZhETF, 71 (1976), 203

[7] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov: Metod obratnoi zadachi, Nauka, M., 1980 | MR

[8] V. S. Buslaev, V. V. Sukhanov, Zap. nauchn. semin. LOMI, 120, 1982, 31 ; 138, 1982, 8 ; Пробл. мат. физ., 10 (1982), 70 ; 11 (1986), 78 | MR | MR | MR | Zbl | MR | Zbl

[9] P. Deift, X. Zhou, “Asymptotics for the Painlevé II equation: Announcement and results”, Spectral Scattering Theory and Applications, Proc. of a Conference on Spectral and Scattering Theory (Tokyo Institute of Technology, June 30–July 3, 1992), Adv. Stud. Pure Math., 23, ed. K. Yajima, Kinokuniya Company Ltd, Tokyo, 1994, 17 | MR | Zbl

[10] E. Ya. Khruslov, Pisma v ZhETF, 21:8 (1975), 469

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

[12] E. Ya. Khruslov, V. P. Kotlyarov, “Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations”, Spectral Operator Theory and Related Topics, Adv. Sov. Math., 19, ed. V. A. Marchenko, AMS, Providence, RI, 1994, 129 | MR

[13] A. V. Gurevich, L. P. Pitaevskii, Pisma v ZhETF, 17:5 (1973), 268

[14] E. Ya. Khruslov, Matem. sb., 99:2 (1976), 261 | MR | Zbl

[15] E. Ya. Khruslov, S. Holger, Mat. fizika, analiz, geometriya, 1998, no. 1/2, 49 | MR | Zbl

[16] V. B. Baranetskii, Mat. fizika, analiz, geometriya, 1999, no. 3/4, 199 | MR

[17] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR