The Kolmogorov spectrum of waves on water is a result of cascading energy via four-wave interactions. For this spectrum in the isotropic case, we introduce basic small perturbations and calculate the decrement of their damping depending on the frequency. We confirm the stability of the Kolmogorov spectrum. The calculation results are applicable for analyzing the stability of numerical methods for solving

We consider the modernized Camassa—Holm equation with periodic boundary conditions. The quadratic nonlinearities in this equation differ substantially from the nonlinearities in the classical Camassa—Holm equation but have all its main properties in a certain sense. We study the so-called nonregular solutions, i.e., those that are rapidly oscillating in the spatial variable. We investigate the problem

We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean the presence of reductions of a chain to a system of hyperbolic equations of an arbitrarily high order that are integrable in the Darboux sense. Darboux integrability admits a remarkable algebraic interpretation: the Lie—Rinehart algebras related

Using the exact Hasselmann equation, we study wind-driven deep-water ocean waves in a strait with the wind directed orthogonally to the shore. The strait has “dissipative” shores with no reflection from the shorelines. We show that the evolution of wave turbulence can be divided into two different regimes in time. During the first regime, waves propagate with the wind, and the wind-driven sea can be

Based on numerical simulation of the nonstationary working of a CW superradiant laser with a low-Q combined Fabry-Perot cavity with distributed feedback of waves, we describe a parametric mechanism of self-mode-locking caused by beats of two superradiant modes with a period half that of the cavity round-trip period and supporting the formation of a soliton of the electromagnetic field inside the cavity

We present the concept of an adiabatic limit of Ginzburg—Landau dynamical equations on ℝ1+2 and Seiberg—Witten equations on four-dimensional symplectic manifolds. We show that the Seiberg—Witten equations can be regarded as a complex version of the Ginzburg—Landau equations.

We consider a boundary-value problem for a quasilinear partial differential equation describing tube oscillations under the action of a fluid flow. We show that the well-known Landau—Hopf scenario of transition to turbulence is realized in the considered evolution boundary-value problem with a suitable choice of the governing parameter. To study the problem, we use the theory of infinite-dimensional

For distributed evolutionary dynamical systems of the “reaction-diffusion” and “reaction-diffusion-advection” types, we analyze the behavior of invariant numerical characteristics of the attractor as the diffusion coefficients decrease. We consider the phenomenon of multimode diffusion chaos, one of whose signatures is an increase in the Lyapunov dimensions of the attractor. For several examples, we

We study the dynamics of the generalized Mackey—Glass equation with two delays using the large-parameter method. After a special exponential replacement of variables, a singularly perturbed equation is obtained, for which we construct a meaningful limit object, a differential—difference relay equation with two delays. We prove that the relay equation has a simple periodic solution with one interval

We classify and analyze transformations of three-dimensional dissipative tangle-solitons, i.e., solitons with closed and nonclosed vortex lines, under smooth variations of the system parameters. An example of such a system is a laser medium with fast saturable absorption. We find several scenarios of both reversible and irreversible transformations and discuss the role of the dissipative property of

We study the nonlinear dynamics of localized perturbations of a confined generic boundary-layer shear flow in the framework of the essentially two-dimensional generalization of the intermediate long-wave (2d-ILW) equation. The 2d-ILW equation was originally derived to describe nonlinear evolution of boundary layer perturbations in a fluid confined between two parallel planes. The distance between the

We consider a system of nonlinear autonomous differential equations that describe the orientation of the antiferromagnetic vector in a multiferroic film and find the conditions for the existence of spatially modulated structures of the cycloid type. We investigate the stability of such structures under spatial perturbations.