The Reduction Transformation of the Generalized Variable-coefficient KdV Equation

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Published: 2023-08-05

DOI: 10.56557/ajomcor/2023/v30i38333

Page: 30-38


Yuqing Chen

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

In this manuscript, we have studied the reduction transformation of the generalized variable-coefficient Korteweg-de Vries(KdV) equation by the modifed Clarkson-Kruskal(CK) direct method, and have established the connection between the generalized variable-coefficient KdV equation and the constant- coefficient KdV equation. After complicated calculations, a new transformation is obtained, which transforms the generalized one-dimensional KdV equation with variable-coefficients into the corresponding KdV equation with constant-coefficients. As we know, the new transformation has not been studied in current literature. Based on the transformation, the solution of variable-coefficient KdV equation can be obtained directly through the constant-coefficient KdV equation, and it is helpful to explore the similarity reduction and exact solution of variable-coefficient KdV equation. Furthermore, a special example is given to verify the correctness of the transformation we have proposed.

Keywords: Variable-coefficient KdV equation;, modified CK direct method, reduction transformation, partial differential equation, solution of differential equations, integrable system


How to Cite

Chen, Y., & Liu, S. (2023). The Reduction Transformation of the Generalized Variable-coefficient KdV Equation. Asian Journal of Mathematics and Computer Research, 30(3), 30–38. https://doi.org/10.56557/ajomcor/2023/v30i38333

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References

Zhang Y, Liu J, Wei G. Lax pair, auto-Backlund transformation and conservation law for a generalized

variable-coefficient KdV equation with external-force term. Appl Math Lett. 2015;45:58-63.

Wei G, Gao Y, Hu W, Zhang C. Painlev analysis, auto-Backlund transformation and new analytic

solutions for a generalized variable-coecient Korteweg-de Vries (KdV) equation. The European Physical

Journal B - Condensed Matter and Complex Systems. 2006;53:343-350.

Zhang Y, Li J, Lv Y. The exact solution and integrable properties to the variable-coecient modi ed

Korteweg-de Vries equation. Annals of Physics. 2008;323:3059-3064.

Alaje A.I , Olayiwola M.O, Adedokun K.A. Adedeji J.A, Oladapo A.O. Modi ed homotopy perturbation

method and its application to analytical solitons of fractional-order KortewegCde Vries equation. Beni-Suef University Journal of Basic and Applied Sciences. 2022;11: n. pag.

Olayiwola M.O. The Variational Iteration Method for Solving Linear and Nonlinear Problems that Arise

in Mathematical Physics.Journal of the Nigerian Association of Mathematical Physics. 2016;35:65-72.

Rosa R.D, Gandarias M.L, Bruzn M.S. Symmetries and conservation laws of a fth-order KdV equation

with time-dependent coecients and linear damping. Nonlinear Dynamics. 2016;84:135-141.

Weiss J, Tabor M, Carnevale GF. The Painlev property for partial di erential equations. Journal of

Mathematical Physics. 1983;24:522-526.

Liu H, Bai C, Xin X. Painleve test, complete symmetry classi cations and exact solutions to R-D types of

equations. Commun. Nonlinear Sci. Numer. Simul. 2021;94:105547.

Liu H, Yue C. Lie symmetries, integrable properties and exact solutions to the variable-coecient nonlinear evolution equations. Nonlinear Dynamics. 2017;89:1989-2000.

Qu C, Ji L. Invariant subspaces and conditional Lie-Backlund symmetries of inhomogeneous nonlinear diffusion equations. Science China Mathematics. 2013;56:2187-2203.

Chen and Liu; Asian J. Math. Comp. Res., vol. 30, no. 3, pp.30-38, 2023; Article no.AJOMCOR.11622

Liu H, Sang B, Xin X, Liu X. CK transformations, symmetries, exact solutions and conservation laws of

the generalized variable-coecient KdV types of equations. J Comput Appl Math. 2019;345:127-134.

Nirmala N, Vedan MJ, Baby B. A variable coecient KortewegCde Vries equation: Similarity analysis

and exact solution. II Journal of Mathematical Physics. 1986;27:2644-2646

Olver P. Applications of Lie Groups To Di erential Equations. Springer, New York; 1993.

Wang M, Zhang J, Li X. Decay mode solutions to cylindrical KP equation. Appl Math Lett. 2016;62:29-34.

El-Shiekh R.M. Periodic and solitary wave solutions for a generalized variable-coecient Boiti-Leon-

Pempinlli system. Comput Math Appl. 2017;73:1414-1420.

Clarkson PA, Kruskal MD. New similarity reductions of the Boussinesq equation. Journal of Mathematical Physics. 1989; 30: 2201-2213.

Lou S, Tang X, Lin J. Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method. Journal of Mathematical Physics. 2000;41:8286-8303.

Lou S, Ma H. LETTER TO THE EDITOR: Non-Lie symmetry groups of (2+1)-dimensional nonlinear

systems obtained from a simple direct method. Journal of Physics A. 2005;38:L129:CL137.

Wang HY, Tian Y, Chen H. Non-Lie Symmetry Group and New Exact Solutions for the Two-Dimensional KdV-Burgers Equation. Chinese Physics Letters. 2011;28:020205.

Wang G, Xu T, Liu X. New Explicit olutions of the Fifth-Order KdV Equation with Variable Coecients.

Bull Malays Math Sci Soc. 2012; 37:769-C778.

Khalique CM, Adem KR. Exact solutions of the (2+1 )-dimensional Zakharov-Kuznetsov modi ed equal width equation using Lie group analysis [J]. Math Comput Model. 2011;54:184-189.

Kraenkel RA, Senthilvelan M, Zenchuk AI. Lie symmetry analysis and reductions of a two-dimensional

integrable generalization of the Camassa-CHolm equation [J]. Physics Letters A. 2011;273:183-193.

Gandarias ML, Bruzn MS. Symmetry group analysis and similarity solutions of the CBS equation in (2+1) dimensions [J]. Pamm. 2008;8:10591-10592.

Devi P, Singh K. Classical Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave system. Asian-european Journal of Mathematics. 2020;2150052.