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).
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).
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).
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: