On the Integration of the matrix modified Korteweg-de Vries equation with a self-consistent source

Article Sidebar

PDF
Published: Sep 3, 2019
DOI: https://doi.org/10.5556/j.tkjm.50.2019.3355
Keywords:
Differential Operators Closed sets Inverse spectral problems

Main Article Content

G. U. Urazboev
A. K. Babadjanova

Abstract

In this work we deduce laws of the evolution of the scattering  data for the matrix Zakharov Shabat system with the potential that is the solution of the matrix modied KdV equation with a self consistent source.

Article Details

How to Cite
Urazboev, G. U., & Babadjanova, A. K. . (2019). On the Integration of the matrix modified Korteweg-de Vries equation with a self-consistent source. Tamkang Journal of Mathematics, 50(3), 281–291. https://doi.org/10.5556/j.tkjm.50.2019.3355
Issue
Vol. 50 No. 3 (2019)
Section
Papers
Author Biographies

G. U. Urazboev

Urgench State University, Uzbekistan.

A. K. Babadjanova

Urgench State University, Uzbekistan

References

Gardner, C.S.; Greene, I.M; Kruskal, M.D.; Miura, R.M. Method for solving the Korteveg-deVries equation. Phys. Rev. Lett., 19, 1095-1097 (1967).

Lax, Peter D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467-490 (1968).

Zakharov, V.E.; Shabat, A.B. Exact theory of two-dimensional self-focusing and one-dimensional self modulation on waves in nonlinear media. Sov. Phys. JETP 34(1) , 62 (1972).

Wadati, M. The modified Korteweg-de Vries equation. J. Phys. Soc. Jpn. 34(5), 1289-1296 (1973).

Wadati, M. Generalized matrix form of the inverse scattering method. In: Bullough R.K., Caudrey, P. J.(eds) Solitons. 17, 287-299 (1980)

Francesco, Demontis; Cornelis Van der Mee. Marchenko equations and norm- ing constants of the matrix Zakharov-Shabat system. Operators and matrices. 2(1), 79-113 (2008).

Faddeev, L.D.; Takhtajan, L.A. Hamiltonian methods in the theory of soli- tons. Springer-Verlag (1987).

Dodd, R.K.; Eilbeck, J.S.; Gibbon, J.D.; Morris, H.C. Solitons and nonlinear wave equations. Academic Press, London (1982).

Lamb, George L. Jr. Elements of soliton theory. New York: Wiley (1980). [10] Ablowitz, M.J.; Segur, H. Solitons and the inverse scattering transform. SIAM Studies in Applied Mathematics (1981).

Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249-315 (1974).

Newell, Alan C. Solitons in mathematics and physics. Philadelphia, PA: Society for Industrial and Applied Mathematics (1985).

Mel’nikov, V.K. A direct method for deriving a multi-soliton solution for the problem of interaction of waves on the x,y plane. Commun. Math. Phys. 112, 639-652 (1987).

Mel’nikov, V. K. Integration method of the Korteweg-de Vries equation with a self-consistent source. Phys. Lett. A 133, 493496 (1988).

Mel’nikov, V.K. Integration of the nonlinear Schrodinger equation with a self- consistent source. Commun. Math. Phys. 137, 359-381 (1991).

Leon, J.; Lati_, A. Solution of an initial-boundary value problem for cou- pled nonlinear waves. J. Phys. A 23, 13851403 (1990).

Shchesnovich, V.S.; Doktorov, E.V. Modi_ed Manakov system with self- consistent source. Phys. Lett. A 213, 23-31 (1996).

Lin, Runliang; Zeng, Yunbo; Ma, Wen-Xiu Solving the KdV hierarchy with self-consistent sources by inverse scattering method. Phys. A 291, 287298 (2001).

Zeng, Yunbo; Ma, Wen-Xiu; Lin, Runliang Integration of the soliton hier- archy with self-consistent sources. J. Math. Phys. 41, 5453-5489 (2000).

Zeng, Yunbo; Ma, Wen-Xiu; Shao, Yijun Two binary Darboux transforma- tions for the KdV hierarchy with self-consistent sources. J. Math. Phys. 42, 2113-2128 (2001)

Goncharenko, V.M. Multisoliton solutions of the matrix KdV equation. Theor. Math. Phys. 126, No.1, 81-91 (2001); translation from Teor. Mat. Fiz. 126, No.1, 102-114 (2001).

Bondarenko, N., Freiling, G., Urazboev, G. Integration of the matrix KdV equation with self consistent source. J. Chaos, solitons & Fractals. 49, 21-27 (2013).

Khalilov, F. A.; Khruslov, E. Ya. Matrix generalization of the modified Korteweg-de Vries equation. Inverse Probl. 6, 193-204 (1990).

Olmedilla, E. Inverse scattering transform for general matrix Schrodinger operators and the related symplectic structure. Inverse Probl. 1, 219-236 (1985).