Asymmetric and Moving-Frame Approaches to the 2D and 3D Boussinesq Equations

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 (Adv. Math. 203 (2006), 497-513) and the three-dimensional stratified rotating Boussinesq equations studied by Hsia, Ma and Wang (J. Math. Phys. 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.