In this paper, a generalized (2+1)-dimensional Hirota–Satsuma–Ito (GHSI) equation is investigated using Lie symmetry approach. Infinitesimal generators and symmetry groups of this equation are presented, and the optimal system is given with adjoint representation. Based on the optimal system, some symmetry reductions are performed and some similarity solutions are provided, including soliton solutions and periodic solutions. With Lagrangian, it is shown that the GHSI equation is nonlinearly self-adjoint. By means of the Lie point symmetries and nonlinear self-adjointness, the conservation laws are constructed. Furthermore, some physically meaningful solutions are illustrated graphically with suitable choices of parameters.

中文翻译：

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广义 (2+1) 维 Hirota-Satsuma-Ito 方程的李对称分析、最优系统和守恒定律

在本文中，使用李对称方法研究了广义 (2+1) 维 Hirota-Satsuma-Ito (GHSI) 方程。给出了该方程的无穷小生成元和对称群，并给出了伴随表示的最优系统。在最优系统的基础上，进行了一些对称性约简，并提供了一些相似性解，包括孤子解和周期解。使用拉格朗日算子，可以证明 GHSI 方程是非线性自伴的。利用李点对称性和非线性自伴随性，构造了守恒定律。此外，一些物理上有意义的解决方案通过适当的参数选择以图形方式说明。