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This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and…
The authors of the paper "Two-dimensional third- and fifth-order nonlinear evolution equations for shallow water waves with surface tension" \cite{Fok} claim that they derived the equation which generalizes the KdV equation to two space…
We study the contributions of off-resonant transitions to the dynamics of a system of N multilevel atoms sharing one excitation and interacting with the quantized vector electromagnetic field. The Rotating Wave Approximation significantly…
We study possible Lie and non-classical reductions of multidimensional wave equations and the special classes of possible reduced equations - their symmetries and equivalence classes. Such investigation allows to find many new conditional…
In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water)…
We present experimental results on turbulence generated in thin fluid layers in the presence of a large-scale coherent flow, or a spectral condensate. It is shown that the condensate modifies the third-order velocity moment in a much wider…
We present a model for nonlinear decay of the weak wave in three-dimensional incompressible magnetohydrodynamic (MHD) turbulence. We show that the decay rate is different for parallel and perpendicular waves. We provide a general formula…
The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears…
The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that…
The last decade has seen a significant increase in the number of studies devoted to wave turbulence. Many deal with water waves, as modeling of ocean waves has historically motivated the development of weak turbulence theory, which adresses…
How does one derive models of dynamical feedback effects in multiscale, multiphysics systems such as wave mean flow interaction (WMFI)? We shall address this question for hybrid dynamical systems, whose motion can be expressed as the…
Coupled wave equations are popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial…
Turbulence is ubiquitous in nonequilibrium systems, and it has been noted that even dense granular flows exhibit characteristics that are typical of turbulent flow, such as the power-law energy spectrum. However, studies on the…
Hydrodynamic equations for a one-component plasma are derived as a generalization of the Euler equations to include the effects of the long-range Coulomb interaction. By using a variational principle, these equations self-consistently unify…
Deep ocean acoustics, in the absence of shipping and wildlife, is driven by surface processes. Best understood is the signal generated by non-linear surface wave interactions, the Longuet-Higgins mechanism, which dominates from 0.1 to 10…
In our previous work, numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the…
Theoretical studies on the modulation of unidimensional regular waves over a flat bottom due to a current typically assign an asymmetry between the effects of opposing/following streams on the evolution of major sea variables, such as…
Many supervised machine learning methods have revolutionised the empirical modelling of complex systems. These empirical models, however, are usually "black boxes" and provide only limited physical explanations about the underlying systems.…
We present the derivation of generic equations describing the long gravity waves in incompressible fluid with decaying effect. We show that in this theory the only restriction to the surface deviation is connected with the stability…
We consider stability of regimes of hydromagnetic thermal convection in a rotating horizontal layer with free electrically conducting boundaries, to perturbations involving large spatial and temporal scales. Equations governing the…