Coherent Structures and Higher Dimensional Solitons in Planetary Fluids

用几个具有二维孤立涡或孤立波解的非线性长波方程来研究高维孤子的可能性和由于高维化而导致的孤子方程的完全可积性的破裂.所考虑的问题包括二维定域结构(定域孤子)相互作用的数值模拟，非线性Rossby波动方程偶极涡(Modon)解的理论可达性分析，得到的主要结果如下：1.数值研究了Petviashvili方程单极涡解和偶极涡解相对于矢量和标量非线性项的稳定性.发现只有单极涡解满足Petviashvili方程.2.从数值和理论两方面研究了调制子与非线性Rossby波动方程的相互作用和倾斜调制子的演化.用多极展开法解析地证明了斜模总是结构不稳定的.3.基于二维局域化孤子的可能性三维非线性长波方程，如Zakharov-Kuznetsov方程或正则化长波方程。考虑了不稳定性和耗散对这类非线性色散方程的非保守影响。结果表明，由多尺度摄动得到的控制近似方程不仅适用于行星流体中的各种波动现象，而且适用于液膜流动、多相流、岩浆运动等各种波动现象。

Several nonlinear long wave equations which admit two-dimensional solitary vortex or solitary wave solutions are taken up to investigate the possibility of higher dimensional soliton and the breakdown of complete integrability of soliton equations due to high-dimensionalization.Considered problems are numerical simulations of interactions of two-dimensionally localized structures (localized solitons) , theoretical atability analysis of dipolar vortex (modon) solutions to the nonlinear Rossby wave equation, and the effects of instability and dissipation on the nonlinear dispersive long wave equations.Obtained main results are as follows.1.Stabilities of monopolar and dipolar vortex solutions of the Petviashvili equation are investigated numerically in relation to the vector and scalar nonlinear terms.Only monopolar vortex solution is found to satisfy the Petviashvili equation.2.Interactions of modons to the nonlinear Rossby wave equation and evolutions of slanted modons are investigated both numerically and theoretically.It is shown analytically by means of the muiltipole expansion method that the slanted modons become always structurally unstable.3.The possibility of two-dimensionally localized solitons is investigated based on the two-dimensional nonlinear long wave equations such as the Zakharov-Kuznetsov or the regularized-long-wave equation. The non-conservative effects of instability and dissipation on such non-linear dispersive equations are considered. It is shown that the governing approximate equations derived by the multiple scale perturbation are ubiquitous equations for a variety of wave phenomena not only in planetary fluid but also in liquid film flow, in multi-phase flow, in magma motion etc.