Abstract: In this paper, we consider the Cauchy problem for the modified Korteweg-de Vries equation with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions. To solve the stated problem, the inverse scattering method is used. Lax pairs are found, which will make it possible to apply the inverse scattering method to solve the stated Cauchy problem. Note that in the case under consideration the Dirac operator is not self-adjoint, so the eigenvalues can be multiple. Equations are found for the dynamics of change in time of the scattering data of a non-self-adjoint operator of the Dirac operator with a potential that is a solution of the modified Korteweg-de Vries equation with variable time-dependent coefficients and with a self-consistent source in the class of rapidly decreasing functions. A special case of a modified Korteweg-de Vries equation with time-dependent variable coefficients and a self-consistent source, namely, a loaded modified Korteweg-de Vries equation with a self-consistent source, is considered. Equations are found for the dynamics of change in time of the scattering data of a non-self-adjoint operator of the Dirac operator with a potential that is a solution of the loaded modified Korteweg-de Vries equation with variable coefficients in the class of rapidly decreasing functions. Examples are given to illustrate the application of the obtained results.
Sobirov, Sh. K. and Hoitmetov, U. A. Integration of the Modified Korteweg-De Vries Equation with Time-Dependent Coefficients and with a Self-Consistent Source, Vladikavkaz Math. J., 2023, vol. 25, no. 3, pp. 123-142 (in Russian).
1. Zabusky, N. J. and Kruskal, M. D. Interaction of Solitons in a Collislontess
Plasma and the Recurrence of Initial States, Physical Review Letters,
1965, vol. 15, no. 6. pp. 240-243.
2. Gardner, C. S., Greene, I. M., Kruskal, M. D. and Miura, R. M.
Method for Solving the Korteweg-de Vries Equation,
Physical Review Letters, 1967, vol. 19, pp. 1095-1097.
3. Lax, P. D. Integrals of Nonlinear Equations of Evolution and Solitary Waves,
Communications on Pure and Applied Mathematics, 1968, vol. 21,
no. 5, pp. 467-490. DOI: 10.1002/cpa.3160210503.
4. Zakharov, V. E. and Shabat, A. B. Exact Theory of Two-Dimensional Self-Focusing and
One-Dimensional Self-Modulation of Waves in Nonlinear Media, Journal of Experimental
and Theoretical Physics, 1972, vol. 34, no. 1, pp. 62-69.
5. Wadati, M. The Exact Solution of the Modified Korteweg-de Vries Equation,
Journal of the Physical Society of Japan, 1972, vol. 32, pp. 1681. DOI: 10.1143/JPSJ.32.1681.
6. Khater, A. H., El-Kalaawy, O. H. and Callebaut, D. K. Backlund Transformations
and Exact Solutions for Alfven Solitons in a Relativistic Electron-Positron Plasma,
Physica Scripta, 1998, vol. 58, no. 6, pp. 545-548. DOI: 10.1088/0031-8949/58/6/001.
7. Tappert, F. D. and Varma, C. M. Asymptotic Theory of Self-trapping of Heat Pulses in Solids,
Physical Review Letters, 1970, vol. 25, pp. 1108-1111. DOI: 10.1103/PhysRevLett.25.1108.
8. Mamedov, K. A. Integration of mKdV Equation with a Self-Consistent
Source in the Class of Finite Density Functions in the Case of Moving Eigenvalues,
Russian Mathematics, 2020, vol. 64, pp. 66-78. DOI: 10.3103/S1066369X20100072.
9. Wu, J. and Geng, X. Inverse Scattering Transform and Soliton Classification
of the Coupled Modified Korteweg-de Vries Equation,
Communications in Nonlinear Science and Numerical Simulation,
2017, vol. 53, pp. 83-93. DOI: 10.1016/j.cnsns.2017.03.022.
10. Khasanov, A. B. and Hoitmetov, U. A. On Integration of the Loaded mKdV Equation
in the Class of Rapidly Decreasing Functions, The Bulletin of Irkutsk State University.
Series Mathematics, 2021, vol. 38, pp. 19-35. DOI: 10.26516/1997-7670.2021.38.19.
11. Vaneeva, O. Lie Symmetries and Exact Solutions of Variable Coefficient
mKdV Equations: an Equivalence Based Approach,
Communications in Nonlinear Science and Numerical Simulation,
2012, vol. 17, no. 2, pp. 611-618. DOI: 10.1016/j.cnsns.2011.06.038.
12. Das, S. and Ghosh, D. AKNS Formalism and Exact Solutions of KdV and Modified
KdV Equations with Variable-Coefficients, International Journal of Advanced Research in Mathematics,
2016, vol. 6, pp. 32-41. DOI: 10.18052/www.scipress.com/IJARM.6.32.
13. Zheng, X., Shang, Y. and Huang, Y. Abundant Explicit and Exact Solutions
for the variable Coefficient mKdV Equations,
Hindawi Publishing Corporation Abstract and Applied Analysis,
2013, 7 p., Article ID 109690. DOI: 10.1155/2013/109690.
14. Demontis, F. Exact Solutions of the Modified Korteweg-de Vries Equation,
Theoretical and Mathematical Physics, 2011, vol. 168, no. 1, pp. 886-897.
DOI: 10.1007/s11232-011-0072-4.
15. Zhang, D.-J., Zhao, S.-L., Sun, Y.-Y. and Zhou, J. Solutions to the Modified Korteweg-de Vries Equation,
Reviews in Mathematical Physics, 2014, vol. 26, no. 7, 1430006, 42 p.
DOI: 10.1142/S0129055X14300064.
16. Hirota, R. Exact Solution of the modified Korteweg-de Vries Equation
for Multiple Collisions of Solitons, Journal of the Physical Society of Japan,
1972, vol. 33, pp. 1456-1458. DOI: 10.1143/JPSJ.33.1456.
17. Gesztesy, T., Schweiger, W. and Simon, B. Commutation Methods Applied to the mKdV-Equation,
Transactions of the American Mathematical Society, 1991, vol. 324,
pp. 465-525. DOI: 10.2307/2001730.
18. Pradhan, K. and Panigrahi, P. K. Parametrically Controlling Solitary Wave Dynamics
in the Modified Korteweg-de Vries Equation, Journal of Physics A: Mathematical and General,
2006, vol. 39, pp. 343-348. DOI: 10.1088/0305-4470/39/20/L08.
19. Yan, Z. The Modified KdV Equation with Variable Coefficients:
Exact uni/bi-variable Travelling Wave-like Solutions,
Applied Mathematics and Computation, 2008,
vol. 203, pp. 106-112. DOI: 10.1016/j.amc.2008.04.006.
20. Khasanov, A. B. On the Inverse Problem of Scattering Theory for a System
of Two Non-Self-Adjoint Differential Equations of the First Order,
Doklady akademii nauk SSSR [Soviet Mathematics -- Doklady],
1984, vol. 277, no. 3, pp. 559-562 (in Russian).
21. Ablowitz, M. J. and Segur, H. Solitons and the Inverse Scattering Transform,
Philadelphia, SIAM, 1987, 438 p.
22. Dodd, R., Eilbeck, J., Gibbon, J. and Morris, H. Solitons and Nonlinear Wave Equations,
London at al, Academic Press, 1982, 630 p.
23. Nakhushev, A. M. Equations of Mathematical Biology, Moscow, Visshaya Shkola,
1995, 304 p. (in Russian).
24. Nakhushev, A. M. Loaded Equations and Their Applications,
Differencial'nye uravnenija [Differential Equations],
1983, vol. 19, no. 1, pp. 86-94 (in Russian).
25. Kozhanov, A. I. Nonlinear Loaded Equations and Inverse Problems,
Computational Mathematics and Mathematical Physics,
2004, vol. 44, no. 4, pp. 657-675.
26. Hasanov, A. B. and 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, vol. 47, no. 2,pp. 250-261. DOI: 10.30546/2409-4994.47.2.250.
27. 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, vol. 33, no. 2, pp. 102-114.
DOI: 10.1134/S1055134422020043.
28. Khasanov, A. B. and 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 Mathematics, 2021, vol. 65, no. 7, pp. 43-57.
DOI: 10.3103/S1066369X21070069.
29. Khasanov, A. B. and Hoitmetov, U. A. On Complex-Valued Solutions of the General
Loaded Korteweg-de Vries Equation with a Source, Differential Equations,
2022, vol. 58, no. 3, pp. 381-391. DOI: 10.1134/S0012266122030089.
30. Hoitmetov, U. A. Integration of the Loaded General Korteweg-de Vries Equation
in the Class of Rapidly Decreasing Complex-valued Functions,
Eurasian Mathematical Journal, 2022, vol. 13, no. 2, pp. 43-54.
DOI: 10.32523/2077-9879-2022-13-2-43-54.
31. Babajanov, B. and Abdikarimov, F. The Application of the Functional Variable Method
for Solving the Loaded Non-Linear Evaluation Equations,
Frontiers in Applied Mathematics and Statistics,
2022, vol. 8, 912674. DOI: 10.3389/fams.2022.912674.