Related papers: On the nonintegrability of equations for long- and…
The integrability of a coupled KdV-mKdV system is tested by means of singularity analysis. The true Lax pair associated with this system is obtained by the use of prolongation technique.
Reductions of the KP-Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev-Petviashvili (KP) equation, are studied. Specifically, the…
We establish the convergence of an approximation scheme to a model for aurora type phenomena. The latter, mathematically, means a system describing the short wave-long wave (SW-LW) interactions for compressible magnetohydrodynamic (MHD)…
This survey is focused on two asymptotic models for internal waves, the Benjamin-Ono (BO) and Intermediate Long Wave (ILW) equations that are integrable by inverse scattering techniques (IST). After recalling briefly their derivations we…
It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics --- amplitude, phase, wave number, etc. --- slowly…
In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…
Nonlinear waves in dispersive media can be succeptible to modulational instabilities. We examine a category of scalar equations, with general dispersion and monomial nonlinearity, including a large variety of KdV-like equations. For…
This article is a brief review of the results of studying the collapse of sound waves in media with positive dispersion, which is described in terms of the three-dimensional Kadomtsev-Petviashvili (KP) equation. The KP instability of…
The propagation of the dispersive shock waves (DSWs) is investigated in the cylindrical Gardner (cG) equation, which is obtained by employing a similarity reduction to the two space one time (2+1) dimensional Gardner-Kadomtsev-Petviashvili…
We study the Klein-Gordon-Zakharov system in two spatial dimensions, an important model in plasma physics. For small, smooth, and spatially localized initial data, we establish the global existence of solutions and characterize their sharp…
Two sets of vectors, covariant and orthogonal Lyapunov vectors (CLVs/OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to…
The interaction between two co-propagating electrostatic wavepackets characterized by arbitrary carrier wavenumber is considered. A one-dimensional (1D) non-magnetized plasma model is adopted, consisting of a cold inertial ion fluid…
We review some essential aspects of classically integrable systems. The detailed outline of the lectures consists of: 1. Introduction and motivation, with historical remarks; 2. Liouville theorem and action-angle variables, with examples…
In nonlinear physics, the interactions among solitons are well studied thanks to the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of…
We introduce and systematically investigate the generation of dispersive shock waves, which arise naturally in physical settings such as optical waveguide arrays and superfluids confined within optical lattices. The underlying physically…
The classical equation of motion of a Davydov model in a coherent state approximation is analyzed using the multiple scales method. An exponentially decaying long range interaction (Kac-Baker model) was included. In the first order, the…
We derive amplitude equations for interface dynamics in pattern forming systems with long-range interactions. The basic condition for the applicability of the method developed here is that the bulk equations are linear and solvable by…
The inverse scattering theory is a basic tool to solve linear differential equations and some Partial Differential Equations (PDEs). Using this theory the Korteweg-de Vries (KdV), the family of evolutionary Non Linear Schrodinger (NLS)…
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…
We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of…