## Similarity Reduction of The Variable-coefficient mKdV Equation

Ziling Zhu

School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China.

Shaowei Liu *

School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China.

*Author to whom correspondence should be addressed.

### Abstract

We use the CK direct method to study a class of variable-coefficient mKdV equations, which deviate the variable coefficient differential equations are converted to ordinary differential equations. As far as we know, no researchers have used this method to study the variable-coefficient mKdV equation in the current literature. The classical Lie group method is only suitable for special forms of g(t) , h(t) , but the CK direct method we use is not only suitable for special forms, but also for general Variable-coefficient mKdV equations. Further, in order to compare whether the reduction results obtained using the two methods are consistent, we use the classical Lie group method and the CK direct method to study the variable-coefficient mKdV equation for a particular g(t) , h(t) . Finally, the results are consistent, which also confirms the correctness of CK direct method.

Keywords: The mKdV equation, similarity reduction, CK direct method, classical Lie group method

#### How to Cite

*Asian Journal of Mathematics and Computer Research*,

*30*(3), 39–50. https://doi.org/10.56557/ajomcor/2023/v30i38361

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### References

Ye Caier, Zhang Weiguo. New exact solutions for the generalized mKdV equation with variable coefficients. Applied Mathematical Sciences. 2011;5(75):3715-3721.

Zhang Ping, Luo Feng, Sun Yuhuai. exact solutions of the general variable coefficient Kdv-Burgers equation. Mathematical Physics; 2020.

Xiao-Ling, Gai, Yi-Tian, Gao, De-Xin, Meng, Lei, Wang, Zhi-Yuan, Sun, Xing, Lü, Qian, Feng, Ming- Zhen, Wang, Xin, Yu, Shun-Hui, Zhu. Darboux transformation and soliton solutions for a variablecoefficient modified kortweg-de vries model from fluid mechanics. Ocean Dynamics, and Plasma Mechanics. Communications in Theoretical Physics. 2010;53(4):673.

Yang Fei, Liu Xiqiang. Exact solution of the variable coefficient GKP equation. Journal of Shandong University (Science Edition). 2019;54(2):111-120.

Clarkson, Peter A, Kruskal, Martin D. New similarity reductions of the Boussinesq equation. Journal of Mathematical Physics. 1989;30(10):2201-2213.

Bluman, George W, Cole, Julian D. Similarity methods for differential equations. Springer Science. 2012;13

Olver, Peter J. Applications of Lie groups to differential equations. Mathematical Physics. 1986;107.

Yan Zhi-lian, Liu, Xi-qiang. Symmetry and similarity solutions of variable coefficients generalized Zakharov–Kuznetsov equation. Elsevier. 2006;180(1):288-294.

Xiqiang, Liu. New explicit solutions to the (2+ 1)-dimensional broer-kaup equations. International Academic Publisher. 2004;17(1):1-11.

Yan, Zhi-lian, Liu, Xi-qiang, Wang, Ling. The direct symmetry method and its application in variable coefficients Schrödinger equation. Applied Mathematics and Computation. 2007;187(2):701-707.

Wang Gangwei, Zhang Yingyuan. Exact solution of the KdV-Burgers equation with variable coefficients. Journal of Liaocheng University: Natural Science Edition. 2011;24(2):9-12.

Lü Hailing, Liu Xiqiang, Liu Na. Explicit solutions of the (2+ 1)-dimensional AKNS shallow water wave equation with variable coefficients. Applied Mathematics and Computation. 2010;217(4):1287-1293.

Tian Guichen, Liu Xiqiang. Exact solutions of the general variable coefficient KdV equation with external force term. Chinese Journal of Quantum Electromics. 2005;22(3):339-343.

Rady AS Abdel, Osman ES, Khalfallah Mohammed. Multi soliton solution for the system of Coupled Korteweg-de Vries equations. Applied Mathematics and Computation. 2009;210(1):177-181. 49 Zhu and Liu; Asian J. Math. Comp. Res., vol. 30, no. 3, pp. 39-50, 2023; Article no.AJOMCOR.11664

Zhang, Sheng, Li, Wei, Zheng, Fu, Yu, JH, Ji, Ming, Lau, ZY, Ma, CZ. A generalized f-expansion method and its application to (2+ 1)-dimensional breaking solition equations. International Journal of Nonlinear Science. 2008;5(1):25-32.

Wazwaz Abdul-Majid. Solitons and singular solitons for the Gardner–KP equation. Elsevier. 2008;204(1):162-169.

Johnpillai, Andrew G, Khalique, Chaudry Masood, Biswas, Anjan. Exact solutions of KdV equation with time-dependent coefficients. Applied Mathematics and Computation. 2010;216(10):3114-3119.

Rosenau P, Schwarzmeier JL. On similarity solutions of Boussinesq-type equations. Communications in Theoretical Physics Physics Letters A. 1986;115(3):75-77.

Tajiri, Masayoshi, Nishitani, Toshiyuki. Two-soliton resonant interactions in one spatial dimension: solutions of Boussinesq type equation. The Physical Society of Japan. 1982;51(11):3720-3723.

Ince, Edward L. Ordinary differential equations. Communications in Theoretical Physics Physics Letters A;1956.

Seadawy AR, El-Rashidy K. Water wave solutions of the coupled system Zakharov-Kuznetsov and generalized coupled KdV equations. The Scientific World Journal; 2014.

Its, Alexander R. The Painlevé transcendents as nonlinear special functions. Painlevé Transcendents: Their Asymptotics and Physical Applications. 1992;49-59.

Ovsiannikov, Lev Vasil’evich. Group analysis of differential equations. Painlevé Transcendents: Their Asymptotics and Physical Applications; 2014.

Schwarz, Fritz. Symmetries of differential equations: From Sophus Lie to computer algebra. Siam Review. 1988;30(3):450-481.

Wu, Jian-Wen, He, Jun-Tao, Lin, Ji. Nonlocal symmetries and new interaction waves of the variablecoefficient modified Korteweg–de Vries equation in fluid-filled elastic tubes. The European Physical Journal Plus. 2022;137(7):1-7.

Wei, Peng-Fei, Long, Chun-Xiao, Zhu, Chen, Zhou, Yi-Ting, Yu, Hui-Zhen, Ren, Bo. Soliton molecules, multi-breathers and hybrid solutions in (2+ 1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation. Chaos, Solitons & Fractals. 2022;158:112062.

Zeng, Xiaohua, Wu, Xiling, Liang, Changzhou, Yuan, Chiping, Cai, Jieping. Exact solutions for coupled variable coefficient KdV equation via quadratic Jacobis elliptic function expansion. Chaos, Solitons & Fractals. 2023;15(5):1021.

El-Shiekh, Rehab M, Gaballah, Mahmoud. Lie group analysis and novel solutions for the generalized variable-coefficients Sawada-Kotera equation. Europhysics Letters. 2023;141(3):32003.

Cheng, Chong-Dong, Tian, Bo, Zhou, Tian-Yu, Shen, Yuan. Wronskian solutions and Pfaffianization for a (3+ 1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma. Physics of Fluids. 2023;35(3).