Solitons, breathers, and localized vortices in hydrodynamics

One more classical object of nonlinear physics is the solitary waves (solitons). Their evolution and interaction play a primary role in the development of the nonlinear stage of various wave perturbations. One of the most interesting and illustrative examples are the large-amplitude solitons in nonintegrable hydrodynamic systems. An important result in this research line was the drawing of a complete analogy between the interaction of solitons and the collision of classical particles. A universal approach to the dynamics of soliton ensembles as particles was developed at the IAP RAS to classify various types of solitary waves by the "attraction — repulsion" criterion, determine the necessary conditions for the existence of bound states of solitons (multisolitons), and formulate a general idea of possible modes of motion in infinite trains of solitons (K. A. Gorshkov, L. A. Ostrovsky, and I. A. Soustova).

Spatio-temporal pattern of propagation of an internal-wave breather in a three-layer flow

Concurrent interaction of large-amplitude envelope solitons on the deep water surface within the framework of complete hydrodynamic equations

New classes of localized oscillating wave packets of large amplitude (breathers or envelope solitons) were studied in multilayer hydrodynamic flows; they exist at times much longer than the characteristic period of a packet. The large-amplitude envelope solitons on the deep sea surface found in numerical calculations are now prototypes of the rogue waves abruptly appearing on the ocean surface; these packets as solitons interact without loss of their identity (E. N. Pelinovsky, A. V. Sergeeva, A. V. Slyunyaev, and T. G. Talipova).
A similar behavior is demonstrated by the localized vortex formations on the liquid surface, the so-called Gerstner waves (A. A. Abrashkin). The evolution of a breather perturbation against homogeneous waves is clearly seen in the figure.

Variation in water surface profile (blue line) and in pressure (red line) over the surface in the Gerstner wave

Other important results of the vortex flow research were obtained at the IAP RAS using the Lagrangian approach to the ideal fluid dynamics. A new class of exact solutions of the hydrodynamic equations was found by means of the Lagrangian variables, which describe nonstationary nonuniformly-eddying plane flows, whose individual fluid particles move along epicycloids or hypocycloids (these flows are called the Ptolemaic ones). A description for the dynamics of a single vortex region in the ambient potential flow (Ptolemaic vortex), which generalizes the classical solution for the Kirchhoff vortex, is given. A matrix formulation for the Lagrangian equations of the ideal fluid dynamics, which provides a spatial generalization of the class of Ptolemaic flows, is proposed. The motion of fluid particles in these flows is the sum of three circular rotations of various amplitudes, frequencies, and spatial orientation (A. A. Abrashkin, D. A. Zenkovich, and E. I. Yakubovich).

Shapes of Ptolemaic vortices at various times


The theory for solitons and vortices was recently applied in a rapidly developing field of nonlinear atomic physics dealing with the coherent waves of matter and enabled one to observe the quantum effects on a macroscopic scale. The wakes behind obstacles moving in a homogeneous Bose-Einstein condensate were explored at the IAP RAS (V. A. Mironov and L. A. Smirnov). The following results have been obtained:
• The subsonic mode of the barrier motion is characterized by the presence of a critical velocity (corresponding to the Landau criterion of the superfluidity theory), above which the laminar (pure potential) flow becomes unstable. In this case, pairs of vortices with different-sign topological charges are excited behind the obstacle. These vortex pairs interact with each other and radiate sound waves, which results in the wake turbulization.
• There is a shadow region inside the Mach cone behind low-intensity supersonic barriers. Highly disturbing supersonic barriers leave a wake of an even number of quasi-one-dimensional density gaps (dark solitons) behind them, which lie inside the Mach cone. Due to the development of modulation instability, the wake evolves into the Karman-type vortex street.
• Both the convergence of a vortex — antivortex pair moving in the region of a denser medium and the formation of a soliton of the Kadomtsev — Petviashvili type are analytically shown using an asymptotic procedure.

Density distribution of the Bose-Einstein condensate flowing around a stationary obstacle created by a potential barrier (left). Enlarged images of the condensate density (center) and wave function phase (right) are shown in separate windows