(3 + 1)维Hirota方程的李对称分析与新精确解

Lie Symmetry Analysis and New Exact Solutions of the (3 + 1)-Dimensional Hirota Equation

摘要: 本文旨在深入探究(3 + 1)维Hirota双线性型非线性偏微分方程的对称性质及其精确解的构造机制。研究采用李对称分析方法，首先基于无穷小生成元理论系统推导出该方程所允许的李点对称代数；在此基础上，选取若干典型对称方向，通过构造相似变量实现对方程的有效降维约化；进一步结合行波变换与变量分离等解析技巧，求解所得低维系统，成功获得了两类新型精确解：一类为具有局域结构特征的孤立波解，另一类为包含任意时间函数与积分形式的广义非行波解。结果表明，李对称分析不仅能够揭示高维非线性演化方程的内在不变结构，而且为系统构建其丰富多样的解析解族提供了一条有效且普适的技术路径，对理解复杂非线性波动力学行为具有理论意义。

Abstract: This paper aims to investigate the symmetry properties and construction mechanisms of exact solutions for the (3+1)-dimensional Hirota bilinear-type nonlinear partial differential equation. By employing Lie symmetry analysis, we first systematically derive the Lie point symmetry algebra admitted by the equation based on infinitesimal generator theory. Subsequently, several representative symmetry directions are selected, and corresponding similarity variables are constructed to effectively reduce the original equation to lower-dimensional systems. Further analytical techniques, including traveling wave transformations and variable separation, are applied to solve these reduced systems, yielding two new classes of exact solutions: one featuring localized solitary wave structures, and the other representing generalized non-traveling-wave solutions involving arbitrary time-dependent functions and integral terms. The results demonstrate that Lie symmetry analysis not only reveals the intrinsic invariant structure of high-dimensional nonlinear evolution equations but also provides an effective and universal approach for systematically constructing diverse families of analytical solutions. This work holds theoretical significance for understanding complex nonlinear wave dynamics.