Nonlinear waves for a variable-coefficient modified Kadomtsev&ndash;Petviashvili system in plasma physics and electrodynamics

Guang-Mei Wei; Yu-Xin Song; Tian-Chi Xing; Shu Miao

doi:10.1088/1572-9494/ad782d

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Communications in Theoretical Physics ›› 2025, Vol. 77 ›› Issue (1) : 15003. DOI: 10.1088/1572-9494/ad782d

Mathematical Physics

Nonlinear waves for a variable-coefficient modified Kadomtsev–Petviashvili system in plasma physics and electrodynamics

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Abstract

In this paper, a variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system is investigated by modeling the propagation of electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film and nonlinear waves in plasma physics and electrodynamics. Painlevé analysis is given out, and an auto-Bäcklund transformation is constructed via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, analytic solutions are given, including the solitonic, periodic and rational solutions. Using the Lie symmetry approach, infinitesimal generators and symmetry groups are presented. With the Lagrangian, the vcmKP equation is shown to be nonlinearly self-adjoint. Moreover, conservation laws for the vcmKP equation are derived by means of a general conservation theorem. Besides, the physical characteristics of the influences of the coefficient parameters on the solutions are discussed graphically. Those solutions have comprehensive implications for the propagation of solitary waves in nonuniform backgrounds.

Key words

modified Kadomtsev-Petviashvili equation / Lie symmetry / optimal system / nonlinear self-adjointness / conservation law / symbolic computation

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Guang-Mei Wei, Yu-Xin Song, Tian-Chi Xing, et al. Nonlinear waves for a variable-coefficient modified Kadomtsev–Petviashvili system in plasma physics and electrodynamics[J]. Communications in Theoretical Physics, 2025, 77(1): 15003 https://doi.org/10.1088/1572-9494/ad782d

References

List( Publishing order | Descend order by publishing year | Descend order by cited within ) Chart analysis

Abdelwahed

H G

, El-Shewy

E K

, Abdelrahman

M A E

, El-Rahmanf

A A

2021 Positron nonextensivity contributions on the rational solitonic, periodic, dissipative structures for VCMKP equation described critical plasmas Adv. Space Res. 67 3260 3266

https://doi.org/10.1016/j.asr.2021.02.015

Cited in this article [1]

2	Seadawy A R, El-Rashidy K 2018 Dispersive solitary wave solutions of Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili dynamical equations in unmagnetized dust plasma Results Phys. 8 1216 1222 https://doi.org/10.1016/j.rinp.2018.01.053

3	Veerakumar V, Daniel M 2003 Modified Kadomtsev-Petviashvili (MKP) equation and electromagnetic soliton Math. Comput. Simulation 62 163 169 https://doi.org/10.1016/S0378-4754(02)00176-3 Cited in this article [1]

4	Sun Z Y, Gao Y T, Yu X, Meng X H, Liu Y 2009 Inelastic interactions of the multiple-front waves for the modified Kadomtsev-Petviashvili equation in fluid dynamics, plasma physics and electrodynamics Wave Motion 46 511 521 https://doi.org/10.1016/j.wavemoti.2009.06.014 Cited in this article [2]

5	Grimshaw R, Pelinovsky D, Pelinovsky E, Talipova T 2001 Wave group dynamics in weakly nonlinear long-wave models Physica D 159 35 57 https://doi.org/10.1016/S0167-2789(01)00333-5 Cited in this article [1]

6	Han P F, Zhang Y 2024 Superposition behavior of the lump solutions and multiple mixed function solutions for the (3+1)-dimensional Sharma-Tasso-Olver-like equation Eur. Phys. J. Plus 139 157 https://doi.org/10.1140/epjp/s13360-024-04953-2

7	Wang D S, Shi Y R, Feng W X, Wen L 2017 Dynamical and energetic instabilities of F = 2 spinor Bose-Einstein condensates in an optical lattice Physica D 351-352 30 41 https://doi.org/10.1016/j.physd.2017.04.002 Cited in this article [1]

8	Hirota R 1971 Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons Phys. Rev. Lett. 27 1192 1193 https://doi.org/10.1103/PhysRevLett.27.1192 Cited in this article [1]

9	Matveev V B, Salle M A 1986 Scattering of solitons in the formalism of the Darboux transform J. Math. Sci. 34 1983 1987 https://doi.org/10.1007/BF01095106 Cited in this article [1]

10	Matveev V B, Salle M A 1991 Darboux Transformations and Solitons Springer

11	Xu L, Wang D S, Wen X Y, Jiang Y L 2020 Exotic localized vector waves in a two-component nonlinear wave system J. Nonlinear. Sci. 30 537 564 https://doi.org/10.1007/s00332-019-09581-0 Cited in this article [1]

12	Rogers C, Schief W K 2002 Bäcklund and Darboux Transformation Geometry and Modern Applications in Soliton Theory Cambridge University Press Cited in this article [1]

13	Weiss J 1983 The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative J. Math. Phys. 24 1405 1413 https://doi.org/10.1063/1.525875 Cited in this article [2]

14	Ma W X 2020 Inverse scattering and soliton solutions of nonlocal complex reverse-spacetime mKdV equations J. Geom. Phys. 157 103845 https://doi.org/10.1016/j.geomphys.2020.103845 Cited in this article [1]

15	Wang D S, Deng S, Zhu X D 2022 Direct and inverse scattering problems of the modified Sawada-Kotera equation: Riemann-Hilbert approach Proc. R. Soc. A. 478 20220541 https://doi.org/10.1098/rspa.2022.0541 Cited in this article [1]

16	Aslan I 2011 Comment on: application of Exp-function method for (3+1)-dimensional nonlinear evolution equations Comput. Math. Appl. 61 1700 1703 https://doi.org/10.1016/j.camwa.2011.01.043 Cited in this article [1]

17	Wang M L 1996 Exact solutions for a compound KdV-Burgers equation Phys. Lett. A 213 279 287 https://doi.org/10.1016/0375-9601(96)00103-X Cited in this article [1]

18	Shukri S, Al-Khaled K 2010 The extended tanh method for solving systems of nonlinear wave equations Appl. Math. Comput. 217 1997 2006 https://doi.org/10.1016/j.amc.2010.06.058 Cited in this article [1]

19	Adem A R 2017 Symbolic computation on exact solutions of a coupled Kadomtsev-Petviashvili equation: lie symmetry analysis and extended tanh method Comput. Math. Appl. 74 1897 1902 https://doi.org/10.1016/j.camwa.2017.06.049

20	Wazwaz A M 2005 The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations Appl. Math. Comput. 167 1196 1210 https://doi.org/10.1016/j.amc.2004.08.005 Cited in this article [1]

21	Han P F, Zhang Y 2024 Investigation of shallow water waves near the coast or in lake environments via the KdV-Calogero-Bogoyavlenskii-Schiff equation Chaos. Soliton. Fract. 184 115008 https://doi.org/10.1016/j.chaos.2024.115008 Cited in this article [1]

22	Olver P J 1993 Application of Lie Groups to Differential Equations Springer Cited in this article [4]

23	Bluman G W, Cheviakov A F, Anco S C 2000 Applications of Symmetry Methods to Partial Differential Equations Springer Cited in this article [2]

24	Kametaka Y 1983 On rational similarity solutions of KdV and mKdV equations Proc. Jpn. Acad. 59 407 409 https://doi.org/10.3792/pjaa.59.407 Cited in this article [1]

25	Kevrekidis P G, Khare A, Saxena A, Herring G 2004 On some classes of mKdV periodic solutions J. Phys. A 37 10959 10965 https://doi.org/10.1088/0305-4470/37/45/014 Cited in this article [1]

26	Konopelchenko B G 1982 On the gauge-invariant description of the evolution equations integrable by Gelfand-Dikij spectral problems Phys. Lett. A 92 323 327 https://doi.org/10.1016/0375-9601(82)90900-8 Cited in this article [2]

27	Jimbo M, Miwa T 1983 Solitons and infinite-dimensional Lie algebras Publ. Res. Inst. Math. Sci. 19 943 1001 https://doi.org/10.2977/prims/1195182017

28	Konopelchenko B G, Dubrovsky V G 1992 Inverse spectral transform for the modified Kadomtsev-Petviashvili equation Stud. Appl. Math. 86 219 268 https://doi.org/10.1002/sapm1992863219 Cited in this article [1]

29	Ren B 2015 Interaction solutions for mKP equation with nonlocal symmetry reductions and CTE method Phys. Scr. 90 065206 https://doi.org/10.1088/0031-8949/90/6/065206 Cited in this article [1]

30	Anco S C 2020 Conservation laws and line soliton solutions of a family of modified KP equations Discrete Contin. Dyn. Syst. Ser. S 13 2655 2665 https://doi.org/10.3390/sym12060950 Cited in this article [1]

31	Zhao Z L 2019 Bäcklund transformations, rational solutions and soliton-cnoidal wave solutions of the modified Kadomtsev-Petviashvili equation Appl. Math. Lett. 89 103 110 https://doi.org/10.1016/j.aml.2018.09.016 Cited in this article [1]

32	Ma W X, Bullough R K, Caudrey P J 1997 Graded symmetry algebras of time-dependent evolution equations and application to the modified KP equations J. Nonlinear Math. Phys. 4 293 309 https://doi.org/10.2991/jnmp.1997.4.3-4.6 Cited in this article [1]

33	Deng X, Chen K, Zhang D J 2017 Lax triad approach to symmetries of scalar modified Kadomtsev-Petviashvili hierarchy Commun. Theor. Phys. 67 131 136 https://doi.org/10.1088/0253-6102/67/2/131 Cited in this article [1]

34	Huang W H, Yu D J, Jin M Z 2003 Some special types of multisoliton solutions of the modified Kadomtsev-Petviashvili equation Commun. Theor. Phys. 40 262 264 https://doi.org/10.1088/0253-6102/40/3/262 Cited in this article [1]

35	Chang J H 2019 Soliton interaction in the modified Kadomtse-Petviashvili-(II) equation Appl. Anal. 98 2589 2603 https://doi.org/10.1080/00036811.2018.1466285 Cited in this article [1]

36	Gao X Y, Guo Y J, Shan W R 2022 Reflecting upon some electromagnetic waves in a ferromagnetic film via a variable-coefficient modified Kadomtsev-Petviashvili system Appl. Math. Lett. 132 108189 https://doi.org/10.1016/j.aml.2022.108189 Cited in this article [2]

37	El-Shiekh R M, Gaballah M 2022 Integrability, similarity reductions and solutions for a (3+1)-dimensional modified Kadomtsev-Petviashvili system with variable coefficients Partial. Differ. Equ. Appl. Math. 6 100408 https://doi.org/10.1016/j.padiff.2022.100408 Cited in this article [1]

38	Zhang Y P, Wang J Y, Wei G M 2015 The Painlevé property, Bäcklund transformation, Lax pair and new analytic solutions of a variable-coefficient KdV equation from fluids and plasmas Phys. Scr. 90 065203 https://doi.org/10.1088/0031-8949/90/6/065203 Cited in this article [1]

39	Luo X Y, Chen Y 2016 Darboux transformation and N-soliton solution for extended form of modified Kadomtsev-Petviashvili equation with variable-coefficient Commun. Theor. Phys. 66 179 188 https://doi.org/10.1088/0253-6102/66/2/179 Cited in this article [1]

40	Ur Rehman H, Tehseen M, Ashraf H, Awan A U, Ali M R 2024 Unveiling dynamic solitons in the (2+1)-dimensional Kadomtsev-Petviashvili equation: insights from fluids and plasma Partial. Differ. Equ. Appl. Math. 9 100633 https://doi.org/10.1016/j.padiff.2024.100633 Cited in this article [1]

41	Weiss J, Tabor M, Carnevale G 1983 The Painlevé property for partial differential equations J. Math. Phys. 24 522 526 https://doi.org/10.1063/1.525721 Cited in this article [2]

42	Xu G Q 2009 A note on the Painlevé test for nonlinear variable-coefficient PDEs Comput. Phys. Commun. 180 1137 1144 https://doi.org/10.1016/j.cpc.2009.01.019 Cited in this article [1]

43	Li X N, Wei G M, Liang Y Q 2010 Painlevé analysis and new analytic solutions for variable-coefficient Kadomtsev-Petviashvili equation with symbolic computation Appl. Math. Comput. 216 3568 3577 https://doi.org/10.1016/j.amc.2010.05.002

44	Liang Y Q, Wei G M, Li X N 2010 Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation Nonlinear Dyn. 62 195 202 https://doi.org/10.1007/s11071-010-9709-3 Cited in this article [1]

45	Zhou T Y, Tian B 2022 Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical Fiber Appl. Math. Lett. 133 108280 https://doi.org/10.1016/j.aml.2022.108280 Cited in this article [1]

46	Olver P J 1995 Equivalence, Invariants and Symmetry Cambridge University Press Cited in this article [4]

47	Ovsiannikov L V 1982 Analysis of Differential Equations Academic Press

48	Bluman G W, Kumei S 1989 Symmetries and Differential Equations Springer

49	Wei G M, Lu Y L, Xie Y Q, Zheng W X 2018 Lie symmetry analysis and conservation law of variable-coefficient Davey-Stewartson equation Comput. Math. Appl. 75 3420 3430 https://doi.org/10.1016/j.camwa.2018.02.008 Cited in this article [1]

50	Guan S N, Wei G M, Li Q 2021 Lie symmetry analysis, optimal system and conservation law of a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation Mod. Phys. Lett. B 35 2150515 https://doi.org/10.1142/S0217984921505151 Cited in this article [1]

51	Ibragimov N H 2007 A new conservation theorem J. Math. Anal. Appl. 333 311 328 https://doi.org/10.1016/j.jmaa.2006.10.078 Cited in this article [3]

52	Ibragimov N H 2011 Nonlinear self-adjointness and conservation laws J. Phys. A. Math. Theor. 44 432002 https://doi.org/10.1088/1751-8113/44/43/432002 Cited in this article [4]