Vortex crystals and non-existence of non-axisymmetric solitary waves in the Flierl-Petviashvili equation (Q1809534)

The authors consider a known (Flierl-Petviashivili) equation for the streamfunction of a two-dimensional large-scale geophysical flow, \({\partial\over\partial t}(\Delta\psi- \psi)+ (1+\psi){\partial\psi\over\partial x}+ J(\psi,\Delta\psi)=0\), where \(J\) is the Jacobian operator, \(J(a,b)= a_xb_y- a_yb_x\). This equation has a well-known solution in the form of a moving axisymmetric vortex. The work reports results of direct numerical simulations which show that steady single-vortex solutions different from the axisymmetric ones do not exist. It is also found that several vortices can form a stable spatially periodic lattice with five or six vortices in the elementary cell, and it is conjectured that other types of vortex lattices exist, too. Additionally, the authors show that quasi-one-dimensional solitons which are described by the KdV equation, are unstable within the framework of the Flierl-Petviashivili equation.