## The Reduction Transformation of the Generalized Variable-coefficient KdV Equation

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

*Asian Journal of Mathematics and Computer Research*,

*30*(3), 30–38. https://doi.org/10.56557/ajomcor/2023/v30i38333

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