The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects

Rong Rong; Hui Liu; Rong Rong; Hui Liu

doi:10.3934/math.2024062

AIMS Mathematics

2024, Volume 9, Issue 1: 1248-1272. doi: 10.3934/math.2024062

Previous Article Next Article

Research article

The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects

Rong Rong ¹,
Hui Liu ^{2
,
,}

1.
School of Mathematics and Statistics, Xiamen University of Technology, Xiamen, Fujian 361024, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Received: 07 October 2023 Revised: 28 November 2023 Accepted: 28 November 2023 Published: 07 December 2023
MSC : 35Q53, 35Q35

The Burgers-KdV equation as a highly nonlinear model, is commonly used in weakly nonlinear analysis to describe small but finite amplitude ion-acoustic waves. In this study, we demonstrate that by considering viscous dissipation, we can derive the Burgers-KdV limit from a one-dimensional plasma system by using the Gardner-Morikawa transformation. This transformation allows us to obtain both homogeneous and inhomogeneous Burgers-KdV equations, which incorporate dissipative and dispersive terms, for the ionic acoustic system. To analyze the remaining system, we employ the energy method in Sobolev spaces to estimate its behavior. As a result, we are able to capture the Burgers-KdV dynamics over a time interval of order $ O(\varepsilon^{-1}) $, where $ \varepsilon $ represents a small parameter.
- Burgers-KdV,
- plasma,
- viscous dissipation,
- Gardner-Morikawa transformation
Citation: Rong Rong, Hui Liu. The Burgers-KdV limit in one-dimensional plasma with viscous dissipation: A study of dispersion and dissipation effects[J]. AIMS Mathematics, 2024, 9(1): 1248-1272. doi: 10.3934/math.2024062

Related Papers:

Abstract

The Burgers-KdV equation as a highly nonlinear model, is commonly used in weakly nonlinear analysis to describe small but finite amplitude ion-acoustic waves. In this study, we demonstrate that by considering viscous dissipation, we can derive the Burgers-KdV limit from a one-dimensional plasma system by using the Gardner-Morikawa transformation. This transformation allows us to obtain both homogeneous and inhomogeneous Burgers-KdV equations, which incorporate dissipative and dispersive terms, for the ionic acoustic system. To analyze the remaining system, we employ the energy method in Sobolev spaces to estimate its behavior. As a result, we are able to capture the Burgers-KdV dynamics over a time interval of order $ O(\varepsilon^{-1}) $, where $ \varepsilon $ represents a small parameter.

References

[1]	M. Qayyum, E. Ahmad, S. Afzal, S. Acharya, Soliton solutions of generalized third order time-fractional KdV models using extended He-Laplace algorithm, Complexity, 2022 (2022), 2174806. https://doi.org/10.1155/2022/2174806 doi: 10.1155/2022/2174806
[2]	D. Benney, Long waves on liquid films, J. Math. Phys., 45 (1966), 150–155. https://doi.org/10.1002/sapm1966451150 doi: 10.1002/sapm1966451150
[3]	R. Johnson, Shallow water waves on a viscous fluid the undular bore, Phys. Fluids, 15 (1972) 1693–1699. https://doi.org/10.1063/1.1693764 doi: 10.1063/1.1693764
[4]	L. Wijngaarden, On the motion of gas bubbles in a perfect fluid, Arch. Mech., 34 (1982), 343–349.
[5]	R. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42 (1970) 49–60. https://doi.org/10.1017/S0022112070001064 doi: 10.1017/S0022112070001064
[6]	T. Yatabe, T. Kanagawa, T. Ayukai, Theoretical elucidation of effect of drag force and translation of bubble on weakly nonlinear pressure waves in bubbly flows, Phys. Fluids, 33 (2021), 033315. https://doi.org/10.1063/5.0033614 doi: 10.1063/5.0033614
[7]	J. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, In: Selected papers of JM Burgers, Dordrecht: Springer, 1995. https://doi.org/10.1007/978-94-011-0195-0_10
[8]	D. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39 (1895), 422–443. https://doi.org/10.1080/14786449508620739 doi: 10.1080/14786449508620739
[9]	D. Gao, Nonplanar ion acoustic solitary waves in an electronegative plasma by damped Korteweg-de Vries-Burgers equation, Chinese J. Phys., 77 (2022), 1789–1795. https://doi.org/10.1016/j.cjph.2021.11.005 doi: 10.1016/j.cjph.2021.11.005
[10]	N. Cao, X. Yin, S. Bai, L. Xu, Breather wave, lump type and interaction solutions for a high dimensional evolution model, Chaos Soliton. Fract., 172 (2023), 113505. https://doi.org/10.1016/j.chaos.2023.113505 doi: 10.1016/j.chaos.2023.113505
[11]	B. Chentouf, A. Guesmia, Well-posedness and stability results for the Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations with infinite memory: A history approach, Nonlinear Anal. Real, 65 (2022), 103508. https://doi.org/10.1016/j.nonrwa.2022.103508 doi: 10.1016/j.nonrwa.2022.103508
[12]	J. Li, K. Liu, Well-posedness of Korteweg-de Vries-Burgers equation on a finite domain, Indian J. Pure Appl. Math., 48 (2017), 91–116. https://doi.org/10.1007/s13226-016-0210-7 doi: 10.1007/s13226-016-0210-7
[13]	R. Duan, H. Zhao, Global stability of strong rarefaction waves for the generalized KdV-Burgers equation, Nonlinear Anal. Theor., 66 (2007), 1100–1117. https://doi.org/10.1016/j.na.2006.01.008 doi: 10.1016/j.na.2006.01.008
[14]	S. EI-Tantawy, A. Salas, M. Alharthi, On the analytical and numerical solutions of the damped nonplanar Shamel Korteweg-de Vries Burgers equation for modeling nonlinear structures in strongly coupled dusty plasmas: Multistage homotopy perturbation method, Phys. Fluids, 33 (2021), 043106. https://doi.org/10.1063/5.0040886 doi: 10.1063/5.0040886
[15]	S. Ivanov, A. Kamchatnov, Formation of dispersive shock waves in evolution of a two-temperature collisionless plasma, Phys. Fluids, 32 (2020), 126115. https://doi.org/10.1063/5.0033455 doi: 10.1063/5.0033455
[16]	L. Molinet, F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Notices, 2002 (2002), 1979–2005. https://doi.org/10.1155/S1073792802112104 doi: 10.1155/S1073792802112104
[17]	T. Dlotko, The generalized Korteweg-de Vries-Burgers equation in $H^2(\mathbb{R})$, Nonlinear Anal. Theor., 74 (2011), 721–732. https://doi.org/10.1016/j.na.2010.08.043 doi: 10.1016/j.na.2010.08.043
[18]	X. Wang, Z. Feng, L. Debnath, D. Y. Gao, The Korteweg-de Vries-Burgers equation and its approximate solution, Int. J. Comput. Math., 85 (2008), 853–863. https://doi.org/10.1080/00207160701411152 doi: 10.1080/00207160701411152
[19]	Z. Feng, R. Knobel, Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities, J. Math. Anal. Appl., 328 (2007), 1435–1450. https://doi.org/10.1016/j.jmaa.2006.05.085 doi: 10.1016/j.jmaa.2006.05.085
[20]	Z. Zhao, L. He, Multiple lump molecules and interaction solutions of the Kadomtsev-Petviashvili I equation, Commun. Theor. Phys., 74 (2022), 105004. https://doi.org/10.1088/1572-9494/ac839c doi: 10.1088/1572-9494/ac839c
[21]	Z. Zhao, L. He, A. Wazwaz, Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves, Chinese Phys. B, 32 (2023), 040501. https://doi.org/10.1088/1674-1056/acb0c1 doi: 10.1088/1674-1056/acb0c1
[22]	Z. Zhao, C. Zhang, Y. Fen, J. Yue, Space-curved resonant solitons and interaction solutions of the (2+1)-dimensional Ito equation, Appl. Math. Lett., 146 (2023), 108799. https://doi.org/10.1016/j.aml.2023.108799 doi: 10.1016/j.aml.2023.108799
[23]	Y. Guo, X. Pu, KdV limit of the Euler-Poisson system, Arch. Rational Mech. Anal., 211 (2014), 673–710. https://doi.org/10.1007/s00205-013-0683-z doi: 10.1007/s00205-013-0683-z
[24]	S. Bai, X. Yin, N. Cao, L. Xu, A high dimensional evolution model and its rogue wave solution, breather solution and mixed solutions, Nonlinear Dyn., 111 (2023), 12479–12494. https://doi.org/10.1007/s11071-023-08467-x doi: 10.1007/s11071-023-08467-x
[25]	X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834–878. https://doi.org/10.48550/arXiv.1204.5435 doi: 10.48550/arXiv.1204.5435
[26]	X. Pu, R. Rong, Zakharov-Kuznetsov-type limit for ion dynamics system with external magnetic field in $\mathbb{R}^3$, Appl. Math. Lett., 115 (2020), 106938. https://doi.org/10.1016/j.aml.2020.106938 doi: 10.1016/j.aml.2020.106938
[27]	L. Yang, X. Pu, Derivation of the burgers equation from the gas dynamics, Commun. Math. Sci., 14 (2016), 671–682. https://doi.org/10.4310/CMS.2016.v14.n3.a4 doi: 10.4310/CMS.2016.v14.n3.a4
[28]	H. Liu, X. Pu, Justification of the NLS approximation for the Euler-Poisson equation, Commun. Math. Phys., 371 (2019), 357–398. https://doi.org/10.1007/s00220-019-03576-4 doi: 10.1007/s00220-019-03576-4
[29]	C. Su, C. Gardner, Korteweg-de Vries equation and generalizations. Ⅲ. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Math. Phys., 10 (1969), 536–539. https://doi.org/10.1063/1.1664873 doi: 10.1063/1.1664873
[30]	L. Song, The Burgers Korteweg-de Vries equation of ionic acoustic waves, Chinese J. space sci., 8 (1988), 57–63. https://doi.org/10.11728/cjss1988.01.053 doi: 10.11728/cjss1988.01.053
[31]	J. Lei, P. Yan, A note on conservation law of evolution equations, Math. Appl., 16 (2003), 75–81.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)