Solutions of a system of two-dimensional Euler equations and stationary structures in an ideal fluid

A system of the Euler equations that describe two-dimensional steady flows of an ideal fluid is considered. This system is reduced to a nonlinear Laplace equation for the stream function. With the use of the Hirota $\tau$-function, solutions of three elliptical equations (sin-Gordon, sinh-Gordon, and Tzitzeica equations) are found. A simple method of deriving solutions in the form of rational expressions in elliptical functions is proposed. The resultant solutions describe sources in a swirled fluid, jet flows, chains of sources and sinks, and vortex structures. It is shown that the fluid flux through a closed curve is quantized in the case of the elliptical sin-Gordon equation.