In this article, the authors study a (2+1)-dimensional MDWW system, which describes the non-linear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The Lie group theoretic approach is employed to find the similarity reductions and analytic solutions of the (2+1)-dimensional MDWW system. The infinitesimal generators for the considered system are obtained under the invariance property of the Lie group of transformations. Later, we construct groups of symmetries and tables for commutation and adjoints. The adjoint table is further used to establish a one-dimensional optimal system of subalgebras. Finally, based on the optimal system, similarity reductions are obtained. A repeated process of similarity reductions reduces the governing system of partial differential equations (PDEs) into systems of ordinary differential equations (ODEs) that generate invariant solutions. Moreover, the dynamical behaviors of the obtained solutions such as multi-soliton, doubly soliton, single soliton, solitary waves, and stationary waves are graphically shown using 3D, 2D, and corresponding contour plots. Thus, physicists and mathematicians can follow complicated physical phenomena more effectively and efficiently using these graphs.

在本文中，作者研究了一个 (2+1) 维的 MDWW 系统，该系统描述了在均匀深度的浅水中沿两个水平方向传播的非线性和色散长重力波。采用李群理论方法来寻找（2+1）维MDWW系统的相似性约简和解析解。所考虑系统的无穷小生成元是在李群变换的不变性下获得的。稍后，我们为交换和伴随构造对称组和表。伴随表进一步用于建立子代数的一维最优系统。最后，基于最优系统，获得相似性降低。相似性降低的重复过程将偏微分方程 (PDE) 的控制系统简化为生成不变解的常微分方程 (ODE) 系统。此外，使用 3D、2D 和相应的等高线图以图形方式显示所获得的解的动力学行为，例如多孤子、双孤子、单孤子、孤立波和驻波。因此，物理学家和数学家可以使用这些图更有效地跟踪复杂的物理现象。