Integration of the extended fifth-order Korteweg-de Vries equation with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions
In this article, the problem of integrating the extended Korteweg-de Vries equation
with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing func-
tions using the inverse scattering method for the self-adjoint Sturm- Liouville operator is studied.
Furthermore, an integration algorithm based on this method is developed, and an illustrative exam-
ple is solved using the proposed algorithm.
1. Gardner C. S.; Greene I. M., Kruskal M. D., Miura R. M. Method for Solving the Korteweg-de Vries
Equation, Phys. Rev. Lett.19 (1967), 1095-1097 .
2. Faddeev L.D. roperties of the S-matrix of one-dimensional equation Schrodinger research Proceedings of
the Institute of Mathematics. them. V.A. Steklova. 73 (1964), 314-336. (In Russian)
3. Lax P.D., Integrals of Nonlinear Equations of Evolution and Solitary Waves Comm. Pure and
Appl.Math.21 (1968), no.5, 467-490.
4. Melnikov V.K. Exact solutions of the Korteweg-de Vries equation with a self- consistent source Phys. Lett.
A (1988) 128, 488-4924.
5. Hoitmetov U.A. Integration of the Hirota equation with time-dependent coefficients Theoretical and Math-
ematical Physics. (2023) 214, no. 1, 24-35.
6. Sobirov Sh.K.; Hoitmetov U.A. Integration of the modified Korteweg-de Vries equation with time-
dependent coefficients and a self-consistent source. Siberian Mathematical Journal. (2024) 65, no. 4,
971-986.
7. Sobirov Sh. Integration of a system of nonlinear integro-differential equations with an additional term in
the class of rapidly decreasing functions. Bull. Inst. Math. (2023) 6, no. 5, 143-156 (In Russian)
8. Hoitmetov U.A. Integration of the sine-gordon equation with time-dependent coefficients and an additional
term. Uzbek Mathematical Journal. (2023) 67, Issue 1, 31-41.
9. Hoitmetov U. On the Cauchy Problem for the mKdV-Sine-Gordon Equation with an Additional Term.
Acta Applicandae Mathematicae. (2023) 184, no.7, 1-15.
10. Michallet H; Barthelemy E. Experimental study of interfacial solitary waves. J. Fluid Mech. (1998) 366,
159-177.
11. Lee C.Y and Beardsley R. C. The generation of long nonlinear internal waves in a weakly stratified shear
flows. J. Geophys. Res. (1974) 79, no.3, 453-457.
12. Kakutani T and Ono H. Weak non-linear hydromagnetic waves in a cold collisionless plasma. J. Phys.
Soc. Jpn. (1969) 26, 1305-1318.
13. Gorshkov K.A.; Ostrovsky L. A and Papko V. V., Interactions and bound states of solitons as classical
particles. Sov. Phys. JETP. (1976) 44,no.2, 306-311.
14. Abramyan L. A and Stepanyants Yu. A., The structure of two-dimensional solitons in media with anoma-
lously small dispersion. Sov. Phys. JETP. (1985) 61, no.5, 963-966.
15. Hunter J. K. and ScheurleJ., Existence of perturbed solitary wave solutions to a model equation for water
waves. Phys. D. (1988) 32, 253-268.
16. Drazin, P.G. and Johnson, R.S. Solitons: An Introduction, Cambridge University Press.(1989) 226.
17. Fucai You; Jiao Zhang, Jianbing Zhang. The N-soliton solutions of the fifth-order KdV equation under
Bargmann constraint.Applied Mathematics and Computation (2010) 217, no.4, 1321-1333.
18. A.M. Wazwaz.Abundant solitons solutions for several forms of the fifth-order KdV equation by using the
tanh method.Applied Mathematics and Computation (2006) 182, no.1, 283-300.
19. Levitan B. Inverse Sturm-Liouville problems , Moscow: Nauka,(1984). 231.(In Russian)
20. Nakhushev A. M. Loaded equations and their applications. Differencialnye Uravnenija [Differential Equa-
tions.] (1983) 19, no.1, 86-94, (in Russian)
21. Hasanov A. B., Hoitmetov U. A. On integration of the loaded Korteweg-de Vries equation in the class
of rapidly decreasing functions. Proceedings of the Institute of Mathematics and Mechanics, National
Academy of Sciences of Azerbaijan. (2021) 47, no.2, 250-261.
22. Hoitmetov 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. Siberian Advances in Mathematics. (2022) 33,
no.2, 102-114.
23. Khasanov A. B.; Hoitmetov U. A. Integration of the General Loaded Korteweg-de Vries Equation with an
Integral Type Source in the Class of Rapidly Decreasing Complex-Valued Functions. Russian Mathemat-
ics. (2021) 65, no.7. 43-57.
24. Hoitmetov U.A.; Khasanov T.G. Integration of the Korteweg-de Vries equation with loaded terms and
a self-consistent source in the class of rapidly decreasing functions. Vestnik Udmurtskogo Universiteta.
Matematika. Mekhanika. Kompyuternye Nauki. (2023) 33, issue 1, 156-170.
Copyright (c) 2026 «ВЕСТНИК НУУз»
Это произведение доступно по лицензии Creative Commons «Attribution-NonCommercial-ShareAlike» («Атрибуция — Некоммерческое использование — На тех же условиях») 4.0 Всемирная.
.jpg)
2.png)
