Various soliton solutions and asymptotic state analysis for the discrete modified Korteweg-de Vries equation. (English) Zbl 1479.35741

Summary: Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized \((m, N - m)\)-fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.


35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35C08 Soliton solutions
35B40 Asymptotic behavior of solutions to PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
82D10 Statistical mechanics of plasmas
78A60 Lasers, masers, optical bistability, nonlinear optics


Full Text: DOI


[1] Peng, W. Q.; Tian, S. F.; Zhang, T. T., Breather waves, high-order rogue waves and their dynamics in the coupled nonlinear Schrödinger equations with alternate signs of nonlinearities, Europhysics letters, 127, 5, article 50005 (2019)
[2] Peng, W. Q.; Tian, S. F.; Zhang, T. T., Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation, Europhysics Letters, 123, 5, article 50005 (2018)
[3] Tamang, J.; Sarkar, K.; Saha, A., Solitary wave solution and dynamic transition of dust ion acoustic waves in a collisional nonextensive dusty plasma with ionization effect, Physica A: Statistical Mechanics and its Applications, 505, 18-34 (2018) · Zbl 07550256
[4] Saha, A.; Banerjee, S., Comment on “multi-dimensional instability of dust acoustic waves in magnetized quantum plasmas with positive or negative dust”, Brazilian Journal of Physics, 51, 4, 1127-1128 (2021)
[5] Tamang, J.; Saha, A., Influence of dust-neutral collisional frequency and nonextensivity on dynamic motion of dust-acoustic waves, Waves in Random and Complex Media, 31, 4, 597-617 (2021) · Zbl 1495.76139
[6] EI-Tantawy, S. A.; Wazwaz, A. M., Anatomy of modified Korteweg-de Vries equation for studying the modulated envelope structures in non-Maxwellian dusty plasmas: freak waves and dark soliton collisions, Physics of Plasmas, 25, 9, article 092105 (2018)
[7] Wazwaz, A. M., New solitons and kink solutions for the Gardner equation, Communications in Nonlinear Science and Numerical Simulation, 12, 8, 1395-1404 (2007) · Zbl 1118.35352
[8] EI-Tantawy, S. A., Nonlinear dynamics of soliton collisions in electronegative plasmas: the phase shifts of the planar KdV- and mkdV-soliton collisions, Chaos, Solitons and Fractals, 93, 162-168 (2016)
[9] EI-Tantawy, S. A.; Carbonaro, P., Nonplanar ion-acoustic solitons collision in \(X e^+- F^--S F_6^-\) and \(A r^+- F^--S F_6^-\) plasmas, Physics Letters A, 380, 18-19, 1627-1634 (2016)
[10] Kaup, D. J., Variational solutions for the discrete nonlinear Schrodinger equation, Mathematics and Computers in Simulation, 69, 3-4, 322-333 (2005) · Zbl 1073.65100
[11] Wadati, M., Transformation theories for nonlinear discrete systems, Progress of Theoretical Physics Supplement, 59, 36-63 (1976)
[12] Toda, M., Theory of Nonlinear Lattices (1989), Berlin: Springer, Berlin · Zbl 0694.70001
[13] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0762.35001
[14] Hirota, R., Exact \(N\)-soliton solution of a nonlinear lumped network equation, Journal of the Physical Society of Japan, 35, 1, 286-288 (1973)
[15] Hu, X. B.; Wu, Y. T., Application of the Hirota bilinear formalism to a new integrable differential- difference equation, Physics Letters A, 246, 6, 523-529 (1998)
[16] Geng, X. G., Darboux transformation of the discrete Ablowitz-Ladik eigenvalue problem, Acta Mathematica Scientia, 9, 1, 21-26 (1989) · Zbl 0693.65091
[17] Xu, X. X., A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation, Applied Mathematics and Computation, 251, 275-283 (2015) · Zbl 1328.37054
[18] Wang, H. T.; Wen, X. Y., Soliton elastic interactions and dynamical analysis of a reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon, Nonlinear Dynamics, 100, 2, 1571-1587 (2020) · Zbl 1459.37061
[19] Xu, T.; Li, H. J.; Zhang, H. J.; Li, M.; Lan, S., Darboux transformation and analytic solutions of the discrete \(PT\)-symmetric nonlocal nonlinear Schrödinger equation, Applied Mathematics Letters, 63, 88-94 (2017) · Zbl 1351.35195
[20] Yu, F. J.; Feng, S., Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with \(4x4\) Lax pairs, Mathematical Methods in the Applied Sciences, 40, 15, 5515-5525 (2017) · Zbl 1384.37093
[21] Yang, Y.; Zhu, Y., Darboux-Backlund transformation, breather and rogue wave solutions for Ablowitz-Ladik equation, Optik, 217, article 164920 (2020)
[22] Zhu, Y.; Yang, Y.; Li, X., Darboux-Backlund transformation, breather and rogue wave solutions for the discrete Hirota equation, Optik, 236, article 166647 (2021)
[23] Wen, X. Y.; Gao, Y. T., Darboux transformation and explicit solutions for discretized modified Korteweg-de Vries lattice equation, Communications in Theoretical Physics, 53, 825-830 (2010) · Zbl 1219.35266
[24] Wen, X. Y.; Yan, Z. Y.; Malomed, B. A., Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: exact solutions and stability, Chaos, 26, 12, article 123110 (2016) · Zbl 1378.35284
[25] Wen, X. Y.; Yan, Z. Y.; Zhang, G. Q., Nonlinear self-dual network equations: modulation instability, interactions of higher order discrete vector rational solitons and dynamical behaviours, Proceedings of The Royal Society A-Mathematical Physical And Engineering Sciences, 476, 2242, article 20200512 (2020) · Zbl 1472.78029
[26] Suris, Y. B., The Problem of Integrable Discretization: Hamiltonian Approach (2003), Basel: Birkhäuser Verlag, Basel · Zbl 1033.37030
[27] Wang, Z.; Zhou, L.; Zhang, H. Q., Solitary solution of discrete mKdV equation by homotopy analysis method, Communications in Theoretical Physics, 49, 1373-1378 (2008) · Zbl 1392.34080
[28] Narita, K., \(N\)-soliton solution of a lattice equation related to the discrete MKdV equation, Journal of Mathematical Analysis And Applications, 381, 2, 963-965 (2011) · Zbl 1226.37050
[29] Zhou, T.; Zhu, Z. N.; He, P., A fifth order semidiscrete mKdV equation, Science China-Mathematics, 56, 1, 123-134 (2013) · Zbl 1339.37064
[30] Xiao, Z.; Li, K.; Zhu, J., Multiple-pole solutions to a semidiscrete modified Korteweg-de Vries equation, Advances in Mathematical Physics, 2019 (2019) · Zbl 1421.35328
[31] Koroglu, C.; Aydin, A., An unconventional finite difference scheme for modified Korteweg-de Vries equation, Advances in Mathematical Physics, 2017 (2017) · Zbl 1404.65093
[32] Liang, J. F.; Wang, X., Investigation of interaction solutions for modified Korteweg-de Vries equation by consistent Riccati expansion method, Mathematical Problems in Engineering, 2019 (2019) · Zbl 1435.35337
[33] Wadati, M., The exact solution of the modified Korteweg-de Vries equation, Journal of the Physical Society of Japan, 32, 6, 1681-1687 (1972)
[34] Wadati, M., The modified Korteweg-de Vries equation, Journal of the Physical Society of Japan, 34, 5, 1289-1296 (1973) · Zbl 1334.35299
[35] Hirota, R., Exact solution of theModifiedKorteweg-de Vries equation for multiple collisions of solitons, Journal of the Physical Society of Japan, 33, 5, 1456-1458 (1972)
[36] Ji, J. L.; Zhu, Z. N., On a nonlocal modified Korteweg-de Vries equation: integrability, Darboux transformation and soliton solutions, Communications in Nonlinear Science and Numerical Simulation, 42, 699-708 (2017) · Zbl 1473.37081
[37] Agrawal, G. P., Nonlinear Fiber Optics (2001), New York: Academic Press, New York
[38] El-Shamy, E. F., Dust-ion-acoustic solitary waves in a hot magnetized dusty plasma with charge fluctuations, Chaos, Solitons Fractals, 25, 3, 665-674 (2005) · Zbl 1084.35070
[39] Shi, Y. R.; Zhang, J.; Yang, H. J.; Duan, W. S., Single soliton of double kinks of the mKdV equation and its stability, Acta Physica Sinica, 59, 11, 7564-7569 (2010) · Zbl 1240.35484
[40] Ablowitz, M. J.; Ladik, J. F., On the solution of a class of nonlinear partial difference equations, Studies in Applied Mathematics, 57, 1, 1-12 (1977) · Zbl 0384.35018
[41] Ablowitz, M. J.; Ladik, J. F., Nonlinear differential-difference equations, Journal of Mathematical Physics, 16, 3, 598-603 (1975) · Zbl 0296.34062
[42] Trefethen, L. N., Spectral Methods in MATLAB (2000), Philadelphia: SIAM, Philadelphia · Zbl 0953.68643
[43] Meng, X. H.; Wen, X. Y.; Piao, L. H.; Wang, D. S., Determinant solutions and asymptotic state analysis for an integrable model of transient stimulated Raman scattering, Optik, 200, article 163348 (2020)
[44] Zhang, C.; Chen, G., The soliton solutions and long-time asymptotic analysis for a general coupled KdV equation, Advances in Mathematical Physics, 2021 (2021) · Zbl 1479.35756
[45] Yuan, C. L.; Wen, X. Y.; Wang, H. T.; Liu, Y. Q., Soliton interactions and their dynamics in a higher-order nonlinear self-dual network equation, Chinese Journal of Physics, 64, 45-53 (2020)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.