Coherent Structures and Higher Dimensional Solitons in Planetary Fluids

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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.