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An initial-boundary value problem on the half-line for the MKdV equation. (English. Russian original) Zbl 0966.35115
Funct. Anal. Appl. 34, No. 1, 52-59 (2000); translation from Funkts. Anal. Prilozh. 34, No. 1, 65-75 (2000).
Summary: The initial-boundary value problem on the half-line for the modified Korteweg-de Vries (MkdV) equation with zero boundary conditions and arbitrary rapidly decaying initial conditions is embedded in the scheme of the inverse scattering method. The corresponding inverse scattering problem is reduced to the Riemann problem on a system of rays in the complex plane.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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[1] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Soliton Theory. Inverse Scattering Method [in Russian], Nauka, Moscow 1980.
[2] A. S. Fokas and A. R. Its, ”An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates,” Teor. Mat. Fiz.,92, No. 3, 386–403 (1992). · Zbl 0802.35133
[3] M. J. Ablowitz and H. J. Segur, ”The inverse scattering problem in a semi-infinite interval,” J. Math. Phys.,16, 1054–1066 (1975). · Zbl 0299.35076
[4] E. K. Sklyanin, ”Boundary conditions for integrable equations,” Funkts. Anal. Prilozhen.,21, No. 2, 86–87 (1987). · Zbl 0643.35093
[5] V. O. Tarasov, ”The integrable initial-boundary value problem on a semiline: nonlinear Schrodinger and sine-Gordon equations,” Inverse Problems,7, No. 3, 435–449 (1991). · Zbl 0732.35089
[6] I. T. Habibullin, ”On integrability of initial-boundary value problems,” Teor. Mat. Fiz.,86, No. 1, 43–51 (1991).
[7] I. T. Habibullin, ”Symmetries of boundary problems,” Phys. Lett. A,178, No. 5–6, 369–375 (1993).
[8] B. Gürel, M. Gürses, and I. Habibullin, ”Boundary value problems for integrable equations compatible with the symmetry algebra,” J. Math. Phys.,36, No. 12, 6809–6821 (1995). · Zbl 0845.35106
[9] V. Adler, B. Gürel, M. Gürses, and I. Habibullin, ”Boundary conditions for integrable equations,” J. Phys. A, Math. Gen.,30, 3505–3513 (1997). · Zbl 0927.35093
[10] B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations, London, 1958. · Zbl 0082.32101
[11] I. M. Krichever, ”An analogue of the d’Alambert formula for the equations of a principal chiral field and the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,253, 288–292 (1980).
[12] M. D. Ramazanov, ”A boundary value problem for a class of differential equations,” Mat. Sb.,64, No. 2, 234–261 (1964).
[13] An Ton Bui, ”Initial boundary value problem for the Korteweg-de Vries equation,” J. Differential Equations,25, No. 3, 288–309 (1977). · Zbl 0372.35070
[14] A. V. Faminskii, ”A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations,” Trudy Mosk. Mat. Obshch.,51, 54–94 (1988).
[15] V. E. Adler, I. T. Habibullin, and A. B. Shabat, ”Boundary value problem for the KdV equation on a half-line,” Teor. Mat. Fiz,110, No. 1, 98–113 (1997). · Zbl 0916.35100
[16] A. B. Shabat, ”The inverse scattering problem,” Differents. Uravn.,15, No. 10, 1824–1835 (1979).
[17] S. Prössdorf, Einige Klassen singularer Gleichugen, Akademie-Verlag, Berlin, 1977.
[18] M. Jimbo and T. Miwa, ”Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II,” Phys. D,2, No. 3, 407–448 (1981). · Zbl 1194.34166
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