Under investigation in this paper is the $(2+1)$ -dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.

中文翻译：

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KdV方程\（（2 + 1）\）维可积耦合的不变解和守恒律的构造

本文正在研究的是KdV方程的（2 + 1）$维可积耦合，它在浅水表面波传播中具有应用。首先，基于李对称性，提出了无穷小生成器和对称性的最优系统。同时，通过tanh方法计算出新的精确解析解。另外，根据易卜拉欣莫夫的方法，建立了保护法律。最后，完成了KdV方程耦合解的目标图。