标题： 调和映射和二维Boussinesq方程 显示英文标题

标题： Harmonic maps and 2D Boussinesq equations

摘要： 在拉格朗日变量的框架下，我们开发了一种方法，利用调和映射理论推导二维 Boussinesq 方程的显式解。 通过重新表述由调和函数描述的流解的特征，我们将问题简化为在复空间中求解一个特定的非线性微分系统。 为了求解这个非线性微分系统，我们引入了 Schwarz 导数和预 Schwarz 导数，并推导了具有相等 Schwarz 导数和预 Schwarz 导数的保向调和映射的性质。 我们的方法得到了包含两种基本经典解类的拉格朗日坐标下的显式解。 : Kirchhoff 的椭圆涡旋 (1876) 和 Gerstner 的重力波 (1809，由 Rankine 在 1863 年重新发现)。 显示英文摘要

摘要： Within the framework of Lagrangian variables, we develop a method for deriving explicit solutions to the 2D Boussinesq equations using harmonic mapping theory. By reformulating the characterization of flow solutions described by harmonic functions, we reduce the problem to solving a particular nonlinear differential system in complex space. To solve this nonlinear differential system, we introduce the Schwarzian and pre-Schwarzian derivatives, and derive the properties of the sense-preserving harmonic mappings with equal Schwarzian and pre-Schwarzian derivatives. Our method yields explicit solutions in Lagrangian coordinates that contain two fundamental classes of classical solutions.: Kirchhoff's elliptical vortex (1876) and Gerstner's gravity wave (1809, rediscovered by Rankine in 1863).