Integration of the modified Korteweg–de Vries equation with time-dependent coefficients and with a self-consistent source

Sh. K. Sobirov   ; U.A. Hoitmetov

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 123-142

Résumé

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.

Bibliographie

[1] Zabusky N. J., Kruskal M. D., “Interaction of solitons in a collislontess plasma and the recurrence of initial states”, Phys. Rev. Lett., 15:6 (1965), 240–243 | DOI | Zbl

[2] 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 | DOI | MR

[3] Lax P. D., “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure and Appl. Math., 21:5 (1968), 467–490 | DOI | MR | Zbl

[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, 34:1 (1972), 62–69 | MR

[5] Wadati M., “The exact solution of the modified Korteweg–de Vries equation”, J. Phys. Soc. Japan, 32 (1972), 1681 | DOI | MR

[6] Khater A. H., El-Kalaawy O. H., Callebaut D. K., “Backlund transformations and exact solutions for Alfven solitons in a relativistic electron–positron plasma”, Physica Scripta, 58:6 (1998), 545–548 | DOI | MR

[7] Tappert F. D., Varma C. M., “Asymptotic theory of self-trapping of heat pulses in solids”, Phys. Rev. Lett., 25 (1970), 1108–1111 | DOI | MR

[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, 64 (2020), 66–78 | DOI | MR | Zbl

[9] Wu J., Geng X., “Inverse scattering transform and soliton classification of the coupled modified Korteweg–de Vries equation”, Communications in Nonlinear Science and Numerical Simulation, 53 (2017), 83–93 | DOI | MR | Zbl

[10] Khasanov A. B., Hoitmetov U. A., “On integration of the loaded mKdV equation in the class of rapidly decreasing functions”, The Bulletin of Irkutsk State University. Ser. Math., 38 (2021), 19–35 | DOI | MR | Zbl

[11] Vaneeva O., “Lie symmetries and exact solutions of variable coefficient mKdV equations: an equivalence based approach”, Communications in Nonlinear Science and Numerical Simulation, 17:2 (2012), 611–618 | DOI | MR | Zbl

[12] Das S., Ghosh D., “AKNS formalism and exact solutions of KdV and modified KdV equations with variable-coefficients”, International Journal of Advanced Research in Mathematics, 6 (2016), 32–41 | DOI

[13] Zheng X., Shang Y., Huang Y., “Abundant Explicit and Exact Solutions for the variable Coefficient mKdV Equations”, Hindawi Publishing Corporation Abstract and Applied Analysis, 2013, 109690, 7 pp. | DOI | MR | Zbl

[14] Demontis, F., “Exact Solutions of the Modified Korteweg–de Vries Equation”, Theoretical and Mathematical Physics, 168:1 (2011), 886–897 | DOI | DOI | MR

[15] Zhang D.-J., Zhao S.-L., Sun Y.-Y., Zhou J., “Solutions to the modified Korteweg–de Vries equation”, Reviews in Math. Phys., 26:7 (2014), 1430006, 42 pp. | DOI | MR | Zbl

[16] Hirota R., “Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons”, J. Phys. Soc. Jpn., 33 (1972), 1456–1458 | DOI

[17] Gesztesy T., Schweiger W., Simon B., “Commutation methods applied to the mKdV-equation”, Trans. Amer. Math. Soc., 324 (1991), 465–525 | DOI | MR | Zbl

[18] Pradhan K., Panigrahi P. K., “Parametrically controlling solitary wave dynamics in the modified Korteweg–de Vries equation”, J. Phys. A: Math. Gen., 39 (2006), 343–348 | DOI | MR

[19] Yan Z., “The modified KdV equation with variable coefficients:Exact uni/bi-variable travelling wave-like solutions”, Applied Mathematics and Computation, 203 (2008), 106–112 | DOI | MR | Zbl

[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”, Soviet Mathematics — Doklady, 277:3 (1984), 559–562 (in Russian) | MR | Zbl

[21] Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1987, 438 pp. | MR

[22] Dodd, R., Eilbeck, J., Gibbon, J. and Morris, H., Solitons and Nonlinear Wave Equations, Academic Press, London at al., 1982, 630 pp. | MR | Zbl

[23] Nakhushev, A. M., Equations of Mathematical Biology, Visshaya Shkola, M., 1995, 304 pp. (in Russian)

[24] Nakhushev, A. M., “Loaded Equations and Their Applications”, Differential Equations, 19:1 (1983), 86–94 (in Russian) | MR | Zbl

[25] Kozhanov, A. I., “Nonlinear Loaded Equations and Inverse Problems”, Computational Mathematics and Mathematical Physics, 44:4 (2004), 657–675 | MR | Zbl

[26] Hasanov A. B., Hoitmetov U. A., “On integration of the loaded Korteweg–de Vries equation in the class of rapidly decreasing functions”, Proc. Inst. Math. Mech. NAS Azer., 47:2 (2021), 250–261 | DOI | MR | Zbl

[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 Adv. Math., 33:2 (2022), 102–114 | DOI | MR

[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, 65:7 (2021), 43–57 | DOI | MR | Zbl

[29] Khasanov A. B., Hoitmetov U. A., “On complex-valued solutions of the general loaded Korteweg–de Vries equation with a source”, Diff. Equat., 58:3 (2022), 381–391 | DOI | MR | Zbl

[30] Hoitmetov U. A., “Integration of the loaded general Korteweg–de Vries equation in the class of rapidly decreasing complex-valued functions”, Eurasian Math. J., 13:2 (2022), 43–54 | DOI | MR

[31] Babajanov B., Abdikarimov F., “The Application of the functional variable method for solving the loaded non-linear evaluation equations”, Front. Appl. Math. Stat., 8 (2022), 912674 | DOI