二维和三维Boussinesq方程的非对称和移动标架方法
Asymmetric and Moving-Frame Approaches to the 2D and 3D Boussinesq Equations
摘要：
Boussinesq非线性偏微分方程组是地球物理流体动力学的基本方程组。本文利用非对称思想和移动标架求解了Chae（{ \it Adv. Math.}）研究的具有部分粘性项的二维Boussinesq方程。{ \bf 203}（2006），497-513）和Hsia，Ma和Wang研究的三维分层旋转Boussinesq方程（{ \it J. Math. Phys.} { \bf 48}（2007），no. 6，06560）。我们得到了一族新的多参数函数的显式精确解。其中许多是可能具有实际意义的周期、拟周期、非周期解。通过傅里叶展开和我们的一些解，可以得到间断解。此外，李点对称性被用来简化我们的论点。
英文摘要：
Boussinesq systems of nonlinear partial differential equations are fundamental equations in geophysical fluid dynamics. In this paper, we use asymmetric ideas and moving frames to solve the two-dimensional Boussinesq equations with partial viscosity terms studied by Chae ({\it Adv. Math.} {\bf 203} (2006), 497-513) and the three-dimensional stratified rotating Boussinesq equations studied by Hsia, Ma and Wang ({\it J. Math. Phys.} {\bf 48} (2007), no. 6, 06560). We obtain new families of explicit exact solutions with multiple parameter functions. Many of them are the periodic, quasi-periodic, aperiodic solutions that may have practical significance. By Fourier expansion and some of our solutions, one can obtain discontinuous solutions. In addition, Lie point symmetries are used to simplify our arguments.
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