Publications

We study entire bounded solutions to the equation ∆ u − u + u 3 = 0 in R2. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unifid way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. It is also applicable for more general equations in any dimension.

We study a breather’s properties within the framework of the modified Korteweg–de Vries (mKdV) model, where cubic nonlinearity is essential. Extrema, moments, and invariants of a breather with dierent parameters have been analyzed. The conditions in which a breather moves in one direction or another has been determined. Two limiting cases have been considered: when a breather has an N-wave shape and can be interpreted as two solitons with dierent polarities, and when a breather contains many oscillations and can be interpreted as an envelope soliton of the nonlinear Schrödinger equation (NLS).

A catalogue of anomalously large waves (rogue or freak waves) occurred in the World Ocean during 2011–2018 reported in mass media sources and scientific literature has been compiled and analyzed. It includes 210 hazardous events caused damages or human losses. The majority of events is based on eyewitness accounts, and as a rule is not confirmed by direct measurements. All collected events divided into deep water cases, shallow water cases and occurrences on the coast (gentle beach or rocks). The following parameters have been determined: date, location, damage, description, reference, and weather conditions. The most dangerous areas in the World Ocean in terms of freak waves are highlighted.

Chaotic foliations generalize Devaney's concept of chaos for         dynamical systems. The property of a foliation to         be chaotic is transversal. The existence problem of chaos for a Cartan foliation         is reduced to the corresponding problem for its holonomy pseudogroup of         local automorphisms of a transversal manifold. Chaotic foliations with transversal Cartan         structures are investigated.          A Cartan $(\Phi,X)$-foliation $(M, F)$ that  admits an Ehresmann connection  is         covered by a locally trivial bundle, and the global holonomy group of $(M, F)$         is defined. In this case, the problem is reduced to the level of         the global holonomy group of the foliation, which is a countable discrete subgroup of the         Lie group of automorphisms of some simply connected Cartan $(\Phi,X)$-manifold.         Several classes of Cartan foliations that cannot be chaotic, are indicated.         Examples of chaotic Cartan foliations are constructed.

We study gradient-like flows with no heteroclinic intersections on an n-dimensional (n ≥ 3) sphere from the point of view of topological conjugacy. We prove that the topological conjugacy class of such a flow is completely determined by the bicolor tree corresponding to the frame of separatrices of codimension 1. We show that for such flows the notions of topological equivalence and topological conjugacy coincide (which is not the case if there are limit cycles and connections.

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations of codimension $q,$ ${q\geq 3.}$ Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

A method of windowed spatiotemporal spectral filtering is proposed to segregate different nonlinear wave components and to calculate the surface of free waves. The dynamic kurtosis (i.e., produced by the free wave component) is shown able to contribute essentially to the abnormally large values of the surface displacement kurtosis, according to the direct numerical simulations of realistic sea waves. In this situation the free wave stochastic dynamics is strongly non-Gaussian, and the kinetic equation for sea surface waves fails. Traces of coherent wave patterns are found in the Fourier transform of the directional irregular sea waves; they may form “jets” in the Fourier domain which strongly violate the classic dispersion relation. 

Feynman formulas are representations of solutions to initial value problems, for some parabolic and Schrödinger equations, by the limits of integrals over finite Cartesian powers of some spaces. Two versions of these formulas which were suggested by Feynman himself are associated with names of Trotter and Chernoff respectively. These formulas can be interpreted as approximations for path integrals over spaces of functions of a real variable; the corresponding representations of the solutions to the said equations are usually known as Feynman-Kac formulas. This work presents some new Feynman type formulas, related to the Chernoff theorem, on Riemannian manifolds. The used manifolds are of boundend geometry which include all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a generally non-compact manifold of bounded geometry. A construction of Chernoff approximations is presented for those Feller semigroups in terms of shift operators. This provides approximations for solutions to initial-value problems for parabolic equations with variable coefficients on the manifold. It also yields the weak convergence of a sequence of random walks on the manifold to the diffusion process associated with the elliptic generator. For parallelizable manifolds this result is applied to the Brownian motion

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.

 

We study topological properties of automorphisms of 4-dimensional torus generatedby integer matrices being symplectic either with respect to the standard symplecticstructure in R4 or w.r.t. a nonstandard symplectic structure generated by an integer skew-symmetric nondegenerate matrix. Such symplectic matrix generates a partially hyperbolic automorphism of the torus, if its eigenvalues are a pair of reals outsidethe unit circle and a complex conjugate pair on the unit circle. The main classifying element is the topology of a foliation generated by unstable (stable) leaves ofthe automorphism. There are two different cases, transitive and decomposable ones.For the first case the foliation into unstable (stable) leaves is transitive, for the second case the foliation itself has a sub-foliation into 2-dimensional tori. For both cases the classification is given.  

The theory of long nonlinear oscillating wave packets (breathers) in a stratified fluid with a small density difference in a gravitational field is developed. The theory is based on the Gardner equation and its modifications, which are fully integrable with modern methods of the nonlinear wave theory. Examples of the breather generation are given and the conditions for their stability are discussed.

In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.

We study a class of scalar differential equations on the circle S1. This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes R+ and R+. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.  

The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrödinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.

 In the present paper we construct an example of 4-dimensional flows on $S^3\times S^1$ whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

In the present paper we prove that frames of one-dimensional separatrices in basins of sources of structurally stable 3-diffeomorhisms with two-dimensional expanding attractor are trivially embedded. This result plays an important part in the classification of such systems. The classification was given by V. Grines and E. Zhuzhoma with assumption that all one-dimensional separatrices are trivially embedded into the ambient manifold but the proof of the assumption was never given. Thus, the present paper is a necessary and nontrivial element of the classification of structurally stable diffeomorphisms with codimension one expanding attractors.

The  existence problem for attractors of foliations with transverse linear connection is investigated. In general foliations with transverse linear connection do not admit attractors. A conditions that implies the existence of a global attractor which is a minimal set, is specified. An application to transversely similar pseudo-Riemannian foliations is obtained. The global structure of transversely similar Riemannian foliations is described. Different examples are constructed.

Currently, a complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9,7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of gradient-like flows (Morse-Smale flows without periodic orbits) was established in [3]. In the classical paper [8], it was proved that in the presence of connections (coincidence of saddle separatrices), the topological equivalence class of a Ω-stable flow splits into a continuum of topological conjugacy classes (has moduli). Obviously, each periodic orbit also generates at least one modulus associated with the period of that orbit. In the present work, it was established that the presence of a cell in a flow bounded by two limit cycles leads to the existence of an infinitely many stability moduli. In addition, a criterion for the topological conjugation of flows on such cells was found.

The Smale surgery on the three-dimensional torus allows one to obtain a so-called DA diffeomorphism from the Anosov automorphism. The nonwandering set of a DA diffeomorphism consists of a single two-dimensional expanding attractor and a finite number of source periodic orbits. As shown by V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, the dynamics of an arbitrary structurally stable 3-diffeomorphism with a two-dimensional expanding attractor generalizes the dynamics of a DA diffeomorphism: such a 3-diffeomorphism exists only on the three-dimensional torus, and the two-dimensional attractor is its unique nontrivial basic set, but its nonwandering set may contain isolated saddle periodic orbits together with source periodic orbits. In the present study, we describe a scenario of a simple transition (through elementary bifurcations) from a structurally stable diffeomorphism of the three-dimensional torus with a two-dimensional expanding attractor to a DA diffeomorphism. A key moment in the construction of the arc is the proof that the closure of the separatrices of boundary periodic points of a nontrivial attractor and of isolated saddle periodic points are tamely embedded. This result demonstrates the fundamental difference of the dynamics of such diffeomorphisms from the dynamics of three-dimensional Morse–Smale diffeomorphisms, in which the closure of the separatrices of saddle periodic points may be wildly embedded.

Pairwise interactions of particle-like waves (such as solitons and breathers) are important elementary processes that play a key role in the formation of the rarefied soliton gas statistics. Such waves appear in different physical systems such as deep water, shallow water waves, internal waves in the stratified ocean, and optical fibers. We study the features of different regimes of collisions between a soliton and a breather in the framework of the focusing modified Korteweg–de Vries equation, where cubic nonlinearity is essential. The relative phase of these structures is an important parameter determining the dynamics of soliton–breather collisions. Two series of experiments with different values of the breather’s and soliton’s relative phases were conducted. The waves’ amplitudes resulting from the interaction of coherent structures depending on their relative phase at the moment of collision were analyzed. Wave field moments, which play a decisive role in the statistics of soliton gases, were determined.