Nonlinear Dispersive Media Research Articles

Painlevé Analysis, Lie Symmetries and Exact Solutions for Variable Coefficients Benjamin—Bona—Mahony—Burger (BBMB) Equation

In this paper, a variable-coefficient Benjamin—Bona—Mahony—Burger (BBMB) equation arising as a mathematical model of propagation of small-amplitude long waves in nonlinear dispersive media is investigated. The integrability of such an equation is studied with Painlevé analysis. The Lie symmetry method is performed for the BBMB equation and then similarity reductions and exact solutions are obtained based on the optimal system method. Furthermore different types of solitary, periodic and kink waves can be seen with the change of variable coefficients.

Observation of rogue wave triplets in water waves

Abstract Doubly-localised breather solutions of the nonlinear Schrodinger equation (NLS) are considered to be appropriate models to describe rogue waves in water waves as well as in other nonlinear dispersive media such as fibre optics. Within the hierarchy of this type of formations, the Peregrine breather (PB) is the lowest-order rational solution. Higher-order solutions of this kind may be understood as a nonlinear superposition of fundamental Peregrine solutions. These superpositions are nontrivial and admit only a fixed well prescribed number of elementary breathers in each higher-order solution. Here, we report first observation of second-order solution which in reality is a triplet of rogue waves.

Time-transformation approach to pulse propagation in nonlinear dispersive media: Inclusion of delayed Raman nonlinearity

We extend our time-transformation technique to include the delayed Raman response and use it to study propagation of ultrashort, few-cycle, optical pulses inside a dispersive nonlinear medium. Our technique deals directly with the electric field associated with an optical pulse and can be applied to pulses of arbitrary widths, as it does not make use of the slowly varying envelope approximation. We apply it to optical pulses containing several optical cycles and launched such that they form a third-order optical soliton. We vary the number of optical cycles within the pulse from 1 to 10 and study how the features such as soliton fission, intrapulse Raman scattering, and dispersive-wave generation depend on pulse width and soliton order. We find that for a fixed soliton order, the Raman-induced frequency shift becomes smaller, while the fraction of energy transferred to the dispersive wave increases, as pulse width is reduced. In the special case of a single-cycle pulse, the most dominant effect is self-steepening and it leads to dramatically different features in both the shape and spectrum of output pulses.

Rogue waves in coupled Hirota systems

Exact rogue wave solutions of the coupled Hirota equations are obtained by using a Darboux dressing transformation. An analysis of the spectral parameter condition shows that our coupled rogue waves, equal or unequal in background height, always exist no matter what parameters have been chosen for the starting plane waves. We demonstrate that, compared with the decoupled ones, the coupled rogue waves can appear in a rather striking way, i.e., one wave exhibits a centrally humped structure, while the other may feature one single hole in the center\char22{}a dark rogue wave structure. It is expected that these unusual rogue wave structures due to the two-wave coupling may help explain the extreme wave events in deep ocean or other nonlinear dispersive media.

Few-cycle optical solitary waves in nonlinear dispersive media

We study the propagation of few-cycle optical solitons in nonlinear media with an anomalous, but otherwise arbitrary, dispersion and a cubic nonlinearity. Our approach does not derive from the slowly varying envelope approximation. The optical field is derived directly from Maxwell's equations under the assumption that generation of the third harmonic is a nonresonant process or at least cannot destroy the pulse prior to inevitable linear damping. The solitary wave solutions are obtained numerically up to nearly single-cycle duration using the spectral renormalization method originally developed for the envelope solitons. The theory explicitly distinguishes contributions between the essential physical effects such as higher-order dispersion, self-steepening, and backscattering, as well as quantifies their influence on ultrashort optical solitons.

Self-phase modulation and frequency generation with few-cycle optical pulses in nonlinear dispersive media

We analyze nonlinear effects associated with the spatiotemporal propagation of few-cycle optical pulses in nonlinear dispersive media, including nonlinearity-induced self-phase modulation, generation of higher harmonics, and the effects of diffraction. First, we discuss the nonlinear equations governing the spatiotemporal propagation of short pulses in dispersive and diffractive nonlinear media and demonstrate a link between the field equation and the cubic nonlinear Schr\"odinger equation employed for describing the evolution of the pulse envelope. Then, for several different regimes of competitive scales, we study the self-action effects of few-cycle pulses and describe, both analytically and numerically, the pulse self-modulation and self-focusing, including the transformation of the spectral density and harmonic generation in the cases of weak and strong dispersion. Finally, we analyze the effect of the beam diffraction on self-action of the Gaussian few-cycle pulses.

Observation of a hierarchy of up to fifth-order rogue waves in a water tank

We present experimental observations of the hierarchy of rational breather solutions of the nonlinear Schrödinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of orders higher than five appeared to be unstable relative to the wave-breaking effect of water waves. Due to the strong influence of the wave breaking and relatively small carrier steepness values of the experiment these results for the higher-order solutions do not directly explain the formation of giant oceanic rogue waves. However, our results are important in understanding the dynamics of rogue water waves and may initiate similar experiments in other nonlinear dispersive media such as fiber optics and plasma physics, where the wave propagation is governed by the NLS.

On the influence of internal degrees of freedom on dispersion in microstructured solids

In the present paper a model describing wave propagation in the nonlinear dispersive media with microstructure is investigated. The model is based on the continuum approach following Mindlin's and Eringer's earlier theories which model a microstructure as a deformable cells in a macrostructure assuming that the deformation gradient is small. A generalized version of the Mindlin model called the Mindlin–Engelbrecht–Pastrone model (MEP) is used. The MEP model is solved numerically using the pseudospectral method and localized initial conditions together with periodic boundary conditions. The main focus of the study is on clarifying the influence of internal degrees of freedom of a microstructure on solutions.

New approach to pulse propagation in nonlinear dispersive optical media

We develop an intuitive approach for studying propagation of optical pulses through nonlinear dispersive media. Our new approach is based on the impulse response of linear systems, but we extend the impulse response function using a self-consistent time-transformation approach so that it can be applied to nonlinear media as well. Numerical calculations based on our new approach show excellent agreement with the generalized nonlinear Schrodinger equation in the specific case of the Kerr nonlinearity in both the normal and anomalous dispersion regimes. An important feature of our approach is that it works directly with the electric field associated with an optical pulse and can be applied to pulses of arbitrary width. Numerical calculations performed using single-cycle optical pulses show that our results agree with those obtained with the finite-difference time-domain technique using considerably more computing resources.

Super Rogue Waves: Observation of a Higher-Order Breather in Water Waves

Super rogue waves with an amplitude of up to 5 times the background value are observed in a water-wave tank for the first time. Nonlinear focusing of the local wave amplitude occurs according to the higher-order breather solution of the nonlinear wave equation. The present result shows that rogue waves can also develop from very calm and apparently safe sea states. We expect the result to have a significant impact on studies of extreme ocean waves and to initiate related studies in other disciplines concerned with waves in nonlinear dispersive media, such as optics, plasma physics, and superfluidity.Received 17 January 2012DOI:https://doi.org/10.1103/PhysRevX.2.011015This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

Optical pulsing in a resonator with a highly dispersive nonlinearity

By introducing an intracavity Doppler shift in a resonator with a highly dispersive nonlinear medium, a train of optical pulses is generated whose features are related to the slow/fast-light response of the medium. The cavity transmission is asymmetric and the pulse shape is modified differently depending on the direction of the Doppler shift, hence, on the sign of the group delay provided by the dispersive process.

On the invariances, conservation laws, and conserved quantities of the damped–driven nonlinear Schrödinger equation

We study the invariance, exact solutions, conservation laws, and double reductions of the nonlinear Schrödinger equation with damping and driving terms. The underlying equation is used to model a variety of resonant phenomena in nonlinear dispersive media, inter alia. For the purpose of our analysis, the complex equation is construed as a system of two real partial differential equations.

Theory and Simulation of Ultrafast Intense Pulse Propagation in Extended Media

The theory of femtosecond pulse propagation in dispersive nonlinear media is reviewed with emphasis on modeling light-matter interactions in femtosecond optical filaments. Discussion of the principles underlying the pulse propagation models is followed by the description of the “standard” light-medium interaction model utilized in the ultrafast nonlinear optics, and open problems are identified across the field.

Ultrashort Optical Pulse Propagation in terms of Analytic Signal

We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a second-order nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schrödinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard split-step solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse.

Transformation of a Gaussian pulse reflecting from a nonlinear medium

The transformation of the envelope of a Gaussian pulse incident on and reflecting from a nonlinear dispersive medium is studied. It is shown that the pulse envelope transforms considerably upon reflection near the optical resonance frequency and the nonlinearity of the medium may appreciably distort the reflected pulse. Away from the resonance frequency, conditions may arise when the shift of the reflected pulse does not lead to the loss of the Gaussian form. In this case, the influence of nonlinearity is insignificant.

Analysis of long-time interaction of perturbations in problems of macrokinetics

We developed an approach for the reduction of systems of nonlinear partial differential equations to the general nonlinear parabolic equation for a complex amplitude of an envelope wave. The equation is complexly derived: by using wave packages, method of many scales, modification of Mandelshtam method and taking into account group velocity of an envelope wave, that is typical for an actual nonlinear dispersive medium. We consider the features of such modeling for a system with nonlinearities occurring in the simplest model of a nonisothermal chemical reaction, in particular, of thermal autoignition and flame propagation.

FDTD Computational Study of Ultra-Narrow TM Non-Paraxial Spatial Soliton Interactions

We consider the interaction between two (1+1)D ultra-narrow optical spatial solitons in a nonlinear dispersive medium using the finite-difference time-domain (FDTD) method for the transverse magnetic (TM) polarization. The model uses the general vector auxiliary differential equation (GVADE) approach to include multiple electric-field components, a Kerr nonlinearity, and multiple-pole Lorentz and Raman dispersive terms. This study is believed to be the first considering narrow soliton interaction dynamics for the TM case using the GVADE FDTD method, and our findings demonstrate the utility of GVADE simulation in the design of soliton-based optical switches.

Time-frequency distribution of interferograms from a frequency comb in dispersive media

We investigate general properties of the interferograms from a frequency comb laser in a non-linear dispersive medium. The focus is on interferograms at large delay distances and in particular on their broadening, the fringe formation and shape. It is observed that at large delay distances the interferograms spread linearly and that its shape is determined by the source spectral profile. It is also shown that each intensity point of the interferogram is formed by the contribution of one dominant stationary frequency. This stationary frequency is seen to vary as a function of the path length difference even within the interferogram. We also show that the contributing stationary frequency remains constant if the evolution of a particular fringe is followed in the successive interferograms found periodically at different path length differences. This can be used to measure very large distances in dispersive media.

Goos–Hänchen shift and time delay in dispersive nonlinear media

We present an analysis of the influence of the Goos–Hänchen effect on tunneling times, group delay and dwell time, of electromagnetic waves propagating through an obstacle made of left-handed metamaterial embedded in a dielectric which exhibits saturable type of nonlinearity. The derived equations show that only the group delay, is affected by the Goos–Hänchen shift without any impact on the dwell time. Besides the reduction of the group delay, the most remarkable result is the possibility for total reduction of the Goos–Hänchen shift for finite incident angles. These phenomena are observable in the frequency region for which metamaterial exhibits negative index of refraction.

Pulse dynamics in dispersive nonlinear medium in presence of traveling refractive-index wave

The dynamics of wave packets propagating in nonlinear waveguides in presence of a traveling refractive-index wave is investigated. It is shown that the regime of self-steepening is possible in these waveguides not only at the trailing edge of the wave packet, but also at its leading edge. Nonreciprocal effects are typically observed when the optical pulse and the wave of optical inhomogeneity are either co- or counterpropagating.