A nonlinear Schrцdinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously
vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function
of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is
shown that the modulational instability criteria for the weakly vortical waves and potential Stokes waves on
deep water coincide. The effect of vorticity manifests itself in a shift of the wavenumber of high-frequency filling.
A special case of Gerstner waves with a zero coefficient at the nonlinear term in the NSE is noted.
Arisingmodulations of surface gravitywaves in a shallow-water resonator under harmonic forcing is discovered in laboratory experiments. Different types of modulations are found. When certain conditions are satisfied (appropriate frequency and sufficient force of excitation), the standing waves become modulated, and the envelopes of standing waves propagate in the channel. Strongly nonlinear numerical simulations of the Euler equations are performed reproducing the modulational regimes observed in the laboratory experiments. The physical mechanism responsible for the occurrence of modulated waves is determined on the basis of the simulations; quantitative estimates aremade with the help of a simplified weakly nonlinear theory. This work was initiated by and performed under the guidance of Prof. A. Ezersky. We dedicate this text to the memory of him.
A class of non-stationary surface gravity waves propagating in the
zonal direction in the equatorial region is described in the f -plane approx-
imation. These waves are described by exact solutions of the equations of
hydrodynamics in Lagrangian formulation and are generalizations of Gerstner
waves. The wave shape and non-uniform pressure distribution on a free sur-
face depend on two arbitrary functions. The trajectories of uid particles are
circumferences. The solutions admit a variable meridional current. The dy-
namics of a single breather on the background of a Gerstner wave is studied as
We develop a method for the application of the Inverse Scattering Technique to the analysis of surface water waves and present here some evidence on its efficiency. The general idea is to interpret nonlinear wave groups in terms of soliton-type structures - envelope solitons in the framework of the integrable nonlinear Schrodinger equation. Such analysis can improve understanding of the nonlinear wave group dynamics and, in particular, could help to elaborate tools for short-term forecasting of dangerous waves in the sea. The technique may also be applied to the problem of the information decoding in soliton based optical transmission lines.
Properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20m and larger depth. For basins of smaller depth, the influence of modulational instability is less probable. By using the rational solutions of the nonlinear Schrodinger equation (breathers), it is shown that the rogue wave packet becomes wider and contains more individual waves in intermediate rather than in deep waters, which is also confirmed by observations.
This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.