Department of Mathematics and Statistics, Universidad Nacional de Colombia, Bogotá, FIZMAKO Research Group, Colombia Sede Manizales.
Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt.
ScientificWorldJournal. 2022 May 11;2022:3240918. doi: 10.1155/2022/3240918. eCollection 2022.
In this paper, some exact traveling wave solutions to the integrable Gardner equation are reported. The ansatz method is devoted for deriving some exact solutions in terms of Jacobi and Weierstrass elliptic functions. The obtained analytic solutions recover the solitary waves, shock waves, and cnoidal waves. Also, the relation between the Jacobi and Weierstrass elliptic functions is obtained. In the second part of this work, we derive some approximate analytic and numeric solutions to the nonintegrable forced damped Gardner equation. For the approximate analytic solutions, the ansatz method is considered. With respect to the numerical solutions, the evolution equation is solved using both the finite different method (FDM) and cubic B-splines method. A comparison between different approximations is reported.
本文报道了可积 Gardner 方程的一些精确行波解。我们采用了假设方法，通过 Jacobi 和 Weierstrass 椭圆函数得到了一些精确解。所得到的解析解可以恢复孤波、冲击波和 cnoidal 波。同时，我们还得到了 Jacobi 和 Weierstrass 椭圆函数之间的关系。在本文的第二部分，我们得到了不可积强迫阻尼 Gardner 方程的一些近似解析和数值解。对于近似解析解，我们采用了假设方法。对于数值解，我们采用有限差分法（FDM）和三次 B 样条方法分别对演化方程进行了求解。报告了不同近似之间的比较。
