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In this paper, we analytically study the two-dimensional unsteady irrotational flow of an ideal incompressible fluid in a half-plane whose boundary is assumed to be a linear sink. It is shown that the nonlinear evolution of perturbations of the initial uniform flow is described by a one-dimensional integro-differential equation, which can be considered as a nonlocal generalization of the Hopf equation. This equation can be reduced to a system of ordinary differential equations (ODEs) in the cases of spatially localized or spatially periodic perturbations of the velocity field. In the first case, ODEs describe the motion of a system of interacting virtual point vortex-sinks/sources outside the flow domain. In the second case, ODEs describe the evolution of a finite number of harmonics of the velocity field distribution; this is possible due to the revealed property of the new equation that the interaction of initial harmonics does not lead to generation of new ones. The revealed reductions made it possible to effectively study the nonlinear evolution of the system, in particular, to describe the effect of nonlinearity on the relaxation of velocity field perturbations. It is shown that nonlinearity can significantly reduce the relaxation rate by more than 1.5 times.

The nonlinear dynamics of the interface between two immiscible dielectric liquids at the regime of suppressed Kelvin-Helmholtz instability by external horizontal electric field is studied theoretically. The initial equations of the fluids motion are reduced to a single weakly nonlinear integro-differential equation that describes the interaction of solitary waves (rational solitons) propagating along the interface. The dynamics of two interacting solitons is regular and integrable; they can combine into a stable wave packet (breather). It is shown that the interaction of three solitons becomes complex and, for a wide rang of initial conditions, chaotic. The numerically obtained Poincaré sections demonstrate the destruction of toroidal trajectories in the phase space during the transition of the system to a chaotic regime of fluid motion. Such a behaviour is consistent with the Kolmogorov-Arnold-Moser theory describing quasi-periodic chaotic motion in Hamiltonian systems. At the developed chaotic state, the system fast loses the information on its initial state; the corresponding estimate for Lyapunov exponent is obtained. From the physical point of view, the chaotic behavior of the system is related with structural instability of the soliton triplet. The triplet can decay into a solitary wave and stable breather consisting of two interacting solitons.