Coherent Structures and Higher Dimensional Solitons in Planetary Fluids

本文研究了几个具有二维孤立涡或孤立波解的非线性长波方程，探讨了高维孤立子的可能性以及孤立子方程的高维化导致的完全可积性的破坏，所考虑的问题是二维局域结构相互作用的数值模拟（局域孤子），非线性Rossby波方程偶极涡旋（modon）解的理论稳定性分析，以及不稳定性和耗散对非线性色散长波方程的影响，得到的主要结果如下：1.单极涡和偶极涡的稳定性本文用数值方法研究了Petviashvili方程的矢量和标量非线性项的解，发现只有单极涡旋解满足Petviashvili方程。2.用数值方法和理论方法研究了非线性Rossby波动方程中模态子的相互作用和斜模态子的演化，用多极展开法解析地表明，斜模态子总是3.从Zakharov-Kuznetsov方程和正则化长波方程出发，研究了二维局域孤子的可能性。考虑了不稳定性和耗散对非线性色散方程的非保守效应。结果表明，由多尺度摄动导出的控制近似方程不仅适用于行星流体，而且适用于液膜流、多相流、岩浆运动等多种波动现象。

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.