Related papers: Mathematical Caricature of Large Waves
We consider the initial value problems of the Kadomtsev-Petviashvili (KP) equation for symmetric V-shape initial waves consisting of two semi-infinite line solitons with the same amplitude. Numerical simulations show that the solutions of…
Wave turbulence and eddy turbulence are the two regimes that we may encounter in nature. The attention of fluid mechanics being mainly focused on incompressible hydrodynamics, it is usually the second regime that is treated in books,…
A family of solitary waves is constructed in Frenkel-Kontorova model and its continuum and quasi-continuum approximations. Each solitary waves is characterised by the number of local maxima in its profile and a relation between external…
We define a mapping of the QCD Balitsky-Kovchegov equation in the diffusive approximation with noise and a generalized coupling allowing a common treatment of the fixed and running QCD couplings. It corresponds to the extension of the…
An expression for the mutual coherence function (MCF) of an electromagnetic beam wave propagating through atmospheric turbulence is derived within the confines of the Rytov approximation. It is shown that both the first and second Rytov…
We present a statistical analysis based on the height and return time probabilities of high amplitude wave events in both focusing and defocusing Manakov systems. We find that analytical rational/semirational solutions, associated with…
In this paper, we study standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2. The Anderson-Gross-Pitaevskii equation is a nonlinear Schr\"odinger equation with a confining potential and a multiplicative spatial…
The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional extention, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly…
The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq…
Reaction-diffusion problems are often described at a macroscopic scale by partial derivative equations of the type of the Fisher or Kolmogorov-Petrovsky-Piscounov equation. These equations have a continuous family of front solutions, each…
In recent papers, denoted by MG24, MG25 in this text, we used the Korteweg-de Vries (KdV) equation and its two-dimensional extension, the Kadomtsev-Petviashvili (KP) equation to describe the evolution of wind-driven water wave packets in…
In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schr\"odinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic "freak waves". We also study the…
Cumulation of shock snd detonation waves was considered. Computations were carried out by use of second-order central-difference scheme. Cumulation of waves in cone region with scales of 1 meter was studied. Pictures of flow in shock and…
Extended shallow water wave equations are derived, using the method of asymptotic expansions, from the Euler (or water wave) equations. These extended models are valid one order beyond the usual weakly nonlinear, long wave approximation,…
We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional,…
The modified Kadomtsev-Petviashvili (mKP) equation is shown in this paper to be decomposable into the first two soliton equations of the 2N-coupled Chen-Lee-Liu and Kaup-Newell hierarchies by respectively nonlinearizing two sets of symmetry…
This gives a survey of our recent studies on soliton solutions of the Kadomtsev-Petviashvili equation with an emphasis on the Mach reflection problem in shallow water.
The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free…
Shock wave theory was first studied for gas dynamics, for which shocks appear as compression waves. A shock wave is characterized as a sharp transition, even discontinuity in the flow. In fact, shocks appear in many different physical…
In these lecture notes we discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green…