Dynamics of complex singularities in fluid dynamics and analytical continuation by rational approximants

A motion of ideal incompressible fluid with a free surface is considered in two-dimensional geometry. A time-dependent conformal mapping of the lower complex half-plane of the auxiliary complex variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid's surface. The fluid dynamics is fully characterized by the motion of the complex singularities in $w$ plane obtained by the analytical continuation into area outside of fluid. We consider both the exact dynamics and the short branch cut approximation of the dynamics with the small parameter being the ratio of the length of the branch cut to the distance between its center and the real line of $w$. The exact dynamics is shown to conserve the infinite number of integrals of motion which are residues of poles in auxiliary complex variables as well as it results in the motion of power law branch points. Fluid dynamics in short branch cut approximation is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement. The numerical continuation is performed into the complex plane using the rational approximants. It allows to compare the dynamics of singularities with the analytical predictions including the existence of multiple integrals of motion. We also analyze the dynamics of singularities and finite time blow up of Constantin–Lax–Majda equation which corresponds to non-potential effective motion of non-viscous fluid with competing convection and vorticity stretching terms. A family of exact solutions is found together with the different types of complex singularities approaching the real line in finite times. These singularities are also recovered by numerical analytical continuation.