Monopolar and dipolar vortex solitons in two space dimensions

Abstract

We compute solitary waves which solve [Delta]u - y2 u - (u + yu2)/c = 0 where c is the phase speed in the x direction and y = 0 is the equator. This equation is a heuristic model for Rossby waves on the "equatorial beta-plane" in geophysical fluid dynamics. For positive c only, there are one-signed solutions ("monopole vortices") which are centered in the middle latitudes. When c [epsilon][-[infinity], -1/3 or c > 0, there are dipoles which have matching vortices of opposite sign in each hemisphere. A third class of solutions is composed of equator-spanning monopoles that are unsymmetric in y. In addition to these families of strict solitary waves, there are also quadrapole vortices which are "weakly non-local solitons" in the sense that they almost meet the usual criterion for solitary waves except for a weak--very weak--radiation to infinity. Orthogonal rational Chebyshev functions and Newton's iteration are used to compute numerical solutions, but four analytic approximations are also derived. Although the equation is only a crude model of geophysical waves, its rich diversity offers a good education in solitary waves in two space dimensions.