Propagation of extremely short electromagnetic pulses in a doubly-resonant medium

Y. Frenkel; I. Gabitov; A. Maimistov; V. Roytburd

Home/Archive/arXiv

arXiv · 0812.4794

Propagation of extremely short electromagnetic pulses in a doubly-resonant medium

Y. Frenkel·I. Gabitov·A. Maimistov·V. Roytburd

View original source ↗Read source document ↗Export metadata ↓

Abstract

Propagation of extremely short electromagnetic pulses in a homogeneous doubly-resonant medium is considered in the framework of the total Maxwell-Duffing-Lorentz model, where the Duffing oscillators (anharmonic oscillators with cubic nonlinearities) represent the dielectric response of the medium, and the Lorentz harmonic oscillators represent the magnetic response. The wave propagation is governed by the one-dimensional Maxwell equations. It is shown that the model possesses a one-parameter family of traveling-wave solutions with the structure of single or multiple humps. Solutions are parametrized by the velocity of propagation. The spectrum of possible velocities is shown to be continuous on a small interval at the lower end of the spectrum; elsewhere the velocities form a discrete set. A correlation between the number of humps and the velocity is established. The traveling-wave solutions are found to be stable with respect to weak perturbations. Numerical simulations demonstrate that the traveling-wave pulses collide in a nearly elastic fashion.

Explore related subjects

nlin.PS nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Y. Frenkel, I. Gabitov, A. Maimistov, V. Roytburd. 2008-12-28. Propagation of extremely short electromagnetic pulses in a doubly-resonant medium. https://arxiv.org/abs/0812.4794

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

General conditions for Turing and wave instabilities in reaction-diffusion systems

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction-diffusion system with $n$ components and a diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on a study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to $\infty$, gives a constructive method for choosing diffusion constants. The results are illustrated on a model for the dispersion of malaria, a 3-component FitzHugh-Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4 components

nlin.PS↗

arXiv2026

Multiprecision computation of bright and dark solitons in the discrete nonlinear Schrödinger equation

We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to $N=65{,}250$ grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.

nlin.PS↗

arXiv2026

Amplitude equations for wave bifurcations in reaction-diffusion systems

A wave bifurcation is the counterpart to a Turing instability in reaction-diffusion systems, but where the critical wavenumber corresponds to a pure imaginary pair rather than a zero temporal eigenvalue. Such bifurcations require at least three components and give rise to patterns that are periodic in both space and time. Depending on boundary conditions, these patterns can comprise either rotating or standing waves. Restricting to systems in one spatial dimension, complete formulae are derived for the evaluation of the coefficients of the weakly nonlinear normal form of the bifurcation up to order five, including those that determine the criticality of both rotating and standing waves. The formulae apply to arbitrary $n$-component systems ($n\geq 3$) and their evaluation is implemented in software which is made available as supplementary material. The theory is illustrated on two different versions of three-component reaction-diffusion models of excitable media that were previously shown to feature super- and subcritical wave instabilities and on a five-component model of two-layer chemical reaction. In each case, two-parameter bifurcation diagrams are produced to illustrate the connection between complex dispersion relations and different types of Hopf, Turing, and wave bifurcations, including the existence of several codimension-two bifurcations.

nlin.PS↗