Elliptical vortex solutions, integrable Ermakov structure, and Lax pair formulation of the compressible Euler equations

The 2+1-dimensional compressible Euler equations are investigated here. A power-type elliptic vortex ansatz is introduced and thereby reduction obtains to an eight-dimensional nonlinear dynamical system. The latter is shown to have an underlying integral Ermakov-Ray-Reid structure of Hamiltonian type. It is of interest to notice that such an integrable Ermakov structure exists not only in the density representations but also in the velocity components. A class of typical elliptical vortex solutions termed pulsrodons corresponding to warm-core eddy theory is isolated and its behavior is simulated. In addition, a Lax pair formulation is constructed and the connection with stationary nonlinear cubic Schrodinger equations is established.