Search for: All records

Abstract The equation for a traveling wave on the boundary of a two‐dimensional droplet of an ideal fluid is derived by using the conformal variables technique for free surface waves. The free surface is subject only to the force of surface tension and the fluid flow is assumed to be potential. We use the canonical Hamiltonian variables discovered and map the lower complex plane to the interior of a fluid droplet conformally. The equations in this form have been originally discovered for infinitely deep water and later adapted to a bounded fluid domain.The new class of solutions satisfies a pseudodifferential equation similar to the Babenko equation for the Stokes wave. We illustrate with numerical solutions of the time‐dependent equations and observe the linear limit of traveling waves in this geometry. 

A potential motion of ideal incompressible fluid with a free surface and infinite depth 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 can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider 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 centre and the real line of w . We found that the fluid dynamics in that 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 moving square root branch points and poles. These solutions involve practical initial conditions resulting in jets and overturning waves. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one. 

Abstract The concept of a primitive potential for the Schrödinger operator on the line was introduced in Dyachenko et al. (2016, Phys. D, 333, 148–156), Zakharov, Dyachenko et al. (2016, Lett. Math. Phys., 106, 731–740) and Zakharov, Zakharov et al. (2016, Phys. Lett. A, 380, 3881–3885). Such a potential is determined by a pair of positive functions on a finite interval, called the dressing functions, which are not uniquely determined by the potential. The potential is constructed by solving a contour problem on the complex plane. In this article, we consider a reduction where the dressing functions are equal. We show that in this case, the resulting potential is symmetric, and describe how to analytically compute the potential as a power series. In addition, we establish that if the dressing functions are both equal to one, then the resulting primitive potential is the elliptic one-gap potential.