Related papers: On the Ergodic Theorem for non-linear wave propaga…
We examine the general question of statistical changes experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In our study that enters within the framework of integrable turbulence, we…
In this article we propose a generalization of the theory of diffusion approximation for random ODE to a nonlinear system of random Schr\"{o}dinger equations. This system arises in the study of pulse propagation in randomly birefringent…
Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we…
The propagation of a wave-packet in a nonlinear disordered medium exhibits interesting dynamics. Here, we present an analysis based on the nonlinear Schr\"odinger equation (Gross-Pitaevskii equation). This problem is directly connected to…
General high-order rogue waves in the nonlinear Schroedinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the…
We present exact rational solution for a modified nonlinear Schr$\ddot{o}$dinger equation that takes into account quintic nonlinearity and nonlinear dispersion corrections to the cubic nonlinearity, which could be used to describe rogue…
These notes provide a student-friendly introduction to the theory of gravitational waves in full, non-linear general relativity (GR). We aim for a balance between physical intuition and mathematical rigor and cover topics such as the…
We consider the propagation of wave packets for a nonlinear Schr\"odinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions.…
The analysis of nonlinear wave equations has experienced a dramatic growth in the last ten years or so. The key factor in this has been the transition from linear analysis, first to the study of bilinear and multilinear wave interactions,…
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic…
The propagation of stable coherent entities of an electromagnetic field in nonlinear media with parameters varying in space can be described in the framework of iterations of nonlinear integral transformations. It is shown that for a set of…
We prove the existence and uniqueness of a family of travelling waves in a degenerate (or singular) quasilinear parabolic problem that may be regarded as a generalization of the semilinear Fisher-Kolmogorov-Petrovski-Piscounov equation for…
In the present work, we explore the possibility of developing rogue waves as exact solutions of some nonlinear dispersive equations, such as the nonlinear Schr\"odinger equation, but also, in a similar vein, the Hirota, Davey-Stewartson,…
The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both $L^2$ spaces and weighted-$L^2$ spaces. As a consequence, an example of a flow admitting a…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
A one-dimensional wave propagation through elastically asymmetric media is investigated. A class of metamaterials possessing an arbitrary elastic asymmetry is proposed. This asymmetry results in different wave speeds of tensile and…
We demonstrate symmetric wave propagations in asymmetric nonlinear quantum systems. By solving the nonlinear Sch\"ordinger equation, we first analytically prove the existence of symmetric transmission in asymmetric systems with a single…
We discuss a signal theorem for charged particle detectors where the finite propagation time of the electromagnetic waves produced by a moving charge cannot be neglected. While the original Ramo-Shockley theorem and related extensions are…
Spectral method related to Lame equation with finite-gap potential is used to study the optical cascading equations. These equations are known not to be integrable by inverse scattering method. Due to "partial integrability" two-gap…
We use Maxwell's equations in a sourceless, inhomogeneous medium with continuous permeability $\mu (\mathbf{r}) $ and permittivity $% \epsilon (\mathbf{r}) $ to study the wave propagation. The general form of the wave equation is derived…