Doubly periodic and multiple pole solutions of the sinh-Poisson equation: Application of reciprocal transformations in subsonic gas dynamics

Vortex patterns associated with the sinh-Poisson equation arise in a remarkable manner as relaxation states of the Navier–Stokes equations. Here, doubly periodic and multiple-pole solutions of the sinh-Poisson equation are generated via the Hirota bilinear operator formalism and exploitation of the phenomenon of coalescence of wave numbers. It is then shown how the multi-parameter reciprocal transformations of gas dynamics may be applied to a seed doubly periodic solution of the sinh-Poisson equation to generate associated periodic vortex structures valid in the subsonic flow of a generalized Kármán–Tsien gas.