Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity

We extend a recently developed Hamiltonian formalism for nonlinear wave interaction processes in spatially periodic dielectric structures to the far-off-resonant regime, and investigate numerically the three-wave resonance conditions in a one-dimensional optical medium with $\chi^{(2)}$ nonlinearity. In particular, we demonstrate that the cascading of nonresonant wave interaction processes generates an effective $\chi^{(3)}$ nonlinear response in these systems. We obtain the corresponding coupling coefficients through appropriate normal form transformations that formally lead to the Zakharov equation for spatially periodic optical media.Comment: 14 pages, 4 figure

Time scales and structures of wave interaction

In this paper we give a general account of Wave Interaction Theory which by now consists of two parts: kinetic wave turbulence theory (WTT), using a statistical description of wave interactions, and the D-model recently introduced in \emph{Kartashova, PRE \textbf{86}: 041129 (2012)} describing interactions of distinct modes. Applying time scale analysis to weakly nonlinear wave systems modeled by the focusing nonlinear Sch\"{o}dinger equation, we give an overview of the structures appearing in Wave Interaction Theory, their time scales and characteristic times. We demonstrate that kinetic cascade and D-cascade are not competing processes but rather two processes taking place at different time scales, at different characteristic levels of nonlinearity and due to different physical mechanisms. Taking surface water waves as an example we show that energy cascades in this system occur at much faster characteristic times than those required by the kinetic WTT but can be described as D-cascades. As D-model has no special pre-requisites, it may be rewarding to re-evaluate existing experiments in other wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc. To appear in EP

Travelling waves in nonlinear diffusion-convection-reaction

The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation. \u

Weak turbulence theory of the non-linear evolution of the ion ring distribution

The nonlinear evolution of an ion ring instability in a low-beta magnetospheric plasma is considered. The evolution of the two-dimensional ring distribution is essentially quasilinear. Ignoring nonlinear processes the time-scale for the quasilinear evolution is the same as for the linear instability 1/t_ql gamma_l. However, when nonlinear processes become important, a new time scale becomes relevant to the wave saturation mechanism. Induced nonlinear scattering of the lower-hybrid waves by plasma electrons is the dominant nonlinearity relevant for plasmas in the inner magnetosphere and typically occurs on the timescale 1/t_ql w(M/m)W/nT, where W is the wave energy density, nT is the thermal energy density of the background plasma, and M/m is the ion to electron mass ratio, which has the consequence that the wave amplitude saturates at a low level, and the timescale for quasilinear relaxation is extended by orders of magnitude

Fluid Models for Kinetic Effects on Coherent Nonlinear Alfven Waves. I. Fundamental Theory

Collisionless regime kinetic models for coherent nonlinear Alfven wave dynamics are studied using fluid moment equations with an approximate closure anzatz. Resonant particle effects are modelled by incorporating an additional term representing dissipation akin to parallel heat conduction. Unlike collisional dissipation, parallel heat conduction is presented by an integral operator. The modified derivative nonlinear Schrodinger equation thus has a spatially nonlocal nonlinear term describing the long-time evolution of the envelope of parallel-propagating Alfven waves, as well. Coefficients in the nonlinear terms are free of the 1/(1-beta) singularity usually encountered in previous analyses, and have very a simple form which clarifies the physical processes governing the large amplitude Alfvenic nonlinear dynamics. The nonlinearity appears via coupling of an Alfvenic mode to a kinetic ion-acoustic mode. Damping of the nonlinear Alfven wave appears via strong Landau damping of the ion-acoustic wave when the electron-to-ion temperature ratio is close to unity. For a (slightly) obliquely propagating wave, there are finite Larmor radius corrections in the dynamical equation. This effect depends on the angle of wave propagation relative to B_0 and vanishes for the limit of strictly parallel propagation. Explicit magnetic perturbation envelope equations amenable to further analysis and numerical solution are obtained. Implications of these models for collisionless shock dynamics are discussed.Comment: 34 pages (including 6 figures

Effects of group velocity and multi-plasmon resonances on the modulation of Langmuir waves in a degenerate plasma

We study the nonlinear wave modulation of Langmuir waves (LWs) in a fully degenerate plasma. Using the Wigner-Moyal equation coupled to the Poisson equation and the multiple scale expansion technique, a modified nonlocal nonlinear Schr{\"{o}}dinger (NLS) equation is derived which governs the evolution of LW envelopes in degenerate plasmas. The nonlocal nonlinearity in the NLS equation appears due to the group velocity and multi-plasmon resonances, i.e., resonances induced by the simultaneous particle absorption of multiple wave quanta. We focus on the regime where the resonant velocity of electrons is larger than the Fermi velocity and thereby the linear Landau damping is forbidden. As a result, the nonlinear wave-particle resonances due to the group velocity and multi-plasmon processes are the dominant mechanisms for wave-particle interaction. It is found that in contrast to classical or semiclassical plasmas, the group velocity resonance does not necessarily give rise the wave damping in the strong quantum regime where $\hbar k\sim mv_{F}$ with $\hbar$ denoting the reduced Planck's constant, $m$ the electron mass and $v_F$ the Fermi velocity, however, the three-plasmon process plays a dominant role in the nonlinear Landau damping of wave envelopes. In this regime, the decay rate of the wave amplitude is also found to be higher compared to that in the modest quantum regime where the multi-plasmon effects are forbidden.Comment: 15 pages, 4 figures; Typos are rectifie

Two dimensional time evolution of beam-plasma instability in the presence of binary collisions

Energetic electrons produced during solar flares are known to be responsible for generating solar type III radio bursts. The radio emission is a byproduct of Langmuir wave generation via beam-plasma interaction and nonlinear wave-wave and wave-particle interaction processes. In addition to type III radio bursts, electrons traveling downwards toward the chromosphere lead to the hard X-ray emission via electron-ion collisions. Recently, the role of Langmuir waves on the X-ray-producing electrons has been identified as important, because Langmuir waves may alter the electron distribution, thereby affecting the X-ray profile. Both Coulomb collisions and wave-particle interactions lead electrons to scattering and energy exchange that necessitates considering the two-dimensional (2D) problem in velocity space. The present paper investigates the influence of binary collisions on the beam-plasma instability development in 2D in order to elucidate the nonlinear dynamics of Langmuir waves and binary collisions. The significance of the present findings in the context of solar physics is discussed