Integration of the Loaded Cordeveg-De Fries Equation in a Class of Fast Decreasing Functions

Abstract

This article is devoted to the integration of the loaded source Korteweg-de Vries equation in the class of rapidly decreasing functions. In this work, the Cauchy problem imposed on the Korteweg-de Vries equation was solved using the inverse problem method of the Sturm-Liouville operator scattering theory. Their Yost solutions are defined and integral Levin images are obtained for them. The givens of the scattering theory were described and some of their necessary properties were given, the Gelfand-Levitan-Marchenko integral equation, which is the main integral equation of the inverse problems of the scattering theory, was derived.

Introduction

Conclusion

This article is devoted to the integration of the Korteweg-de Vries equation with an adapted source load in the class of rapidly decreasing functions. This article provides the necessary information on the exact and inverse problems of the scattering theory for the Sturm-Liouville operator, which are necessary for the following statements. First, the Yost solutions of the Sturum-Liouville operator on the entire axis are defined and integral images are obtained for them, the givens of the scattering theory are described and some of their necessary properties are given, the Gelfand-Levitan equation, which is the main integral equation of the inverse problems of the scattering theory, The Marchenco integral equation was derived. The problem of finding the solution of the Cauchy problem in the class of rapidly decreasing functions, which is applied to the Korteweg-de Vries equation with an adapted source load, is studied. In this case, the method of inverse problems of the scattering theory was used to determine the solution of the Cauchy problem imposed on the Korteweg-de Vries equation with an adapted source load in the class of rapidly decreasing functions. Equations for calculating the evolution of the Sturm-Liouville operator given by the scattering theory have been derived. The algorithm of applying the method of inverse problems of scattering theory is given. An example is shown in order to show the correctness of the obtained results.

References

[1] Gardner C., Greene I., Kruskal M., Miura R. A method for solving the Korteweg-de Vries equation. Phys. Rev. Lett., New York, 19, p. 1095-1098 (1967). [2] Faddeev L.D. Properties of the S-matrix of the one-dimensional Schrödinger equation. Tr. MIANSSSR, 73, 314-336., (1964). [3] Marchenko V.A. Sturm-Liouville operators and their applications, Naukova Dumka, Kyiv, 1977. [4] Levitan B.M. Inverse Sturm-Liouville Problems, Nauka, M.: 1984. [5] Lax P.D. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure and Appl. Math., v. 21. 467-490, (1968). [6] Bhatnagar P. Nonlinear waves in one-dimensional disperse systems. Moscow \"Mir\" 1983. [7] Lam JL Introduction to the theory of solitons. Moscow \"Mir\" 1983. [8] Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevskii L.P. Theory of solitons. Inverse problem method. Moscow. \"World\". 1987. [9] Ablovitz M., Sigur H. Solitons and the inverse problem method. Moscow. \"World\". 1987. [10] Takhtadzhyan L.A., Faddeev L.D. Hamiltonian approach in the theory of solitons. Moscow. The science. (1986). [11] R. Dodd, J. Eilbeck, J. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations. Moscow. \"World\". 1988. [12] Novokshenov V.Yu. Introduction to the theory of solitons. Moscow. Izhevsk. 2002. [13] Mel\'nikov V.K. Integration method of the Korteweg-de Vries equation with a self-consistent source. Phys. Lett. A, 133:9 (1988), p. 493-496. [14] Mel\'nikov V.K. Integration of the Korteweg-de Vries equation with a source. Inverse problems 6:2 (1990), 233-246. [15] Leon J., Latifi A. Solution of an initial-boundary value problem for coupled nonlinear waves. J Phys. A: Math. Gen. 23:8 (1990), 1385-1403. [16] Claude C., Latifi A., Leon J. Nonlinear resonant scattering and plasma instability: an integrable model. J Math. Phys., 23:12 (1991), 3321-3330. [17] Zeng Y. Ma W. X., Lin R. Integration of the solution hierarchy with self-consistent source. J Math. Phys., 41:8 (2000), 5453-5489. [18] Hasanov A.B., Hoitmetov U.A. On integration of the loaded Korteweg-de Vries equation in the class of rapidly decreasing functions. Proceeding of the Institute of Math. And Mechan. National academy of sciences of Azerbaijan. Vol., 47, No. 2, 2021, p. 250-261. [19] Khoitmetov U.A. Integration of the loaded KdV equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions. Mathematical works. .t. 24, No. 2, pp. 181-198 (2021). [20] Khasanov A.B., Matyakubov M.M. Integration of the nonlinear Korteweg-de Vries equation with an additional term. TMF., 203, No. 2 (2020), 192-204. [21] Khasanov A.B., Khasanov T.G. The Cauchy problem for the Korteweg-de Vries equation in the class of periodic infinite-gap functions. Notes of scientific seminars POMI. v. 506, pp. 258-278 (2021). [22] Nakhushev A.M. Equations of mathematical biology. M:. Graduate School. (1985) [23] Kozhanov A.I. Nonlinear loaded equations and inverse problems. J. Comput. Mat. and mat. Phys. 44, 694-716 (2004) [24] Lugovtsov A.A. Propagation of Nonlinear Waves in a Unhomogenous Gas-Liquid Medium. Derivation of the Wave equations Close to Korteweg-de Vries, Applied Mech. and tech. Phys., 50:2 (2009), 188-197. [25] Lugovtsov A.A. Propagation of Nonlinear Waves in a Gas-Liquid Medium. Exact and Approximate Analytical Solutions of Wave Equations. Applied Mech. and tech. Phys., 51:1 (2010), 54-61.

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