Three-wave Interaction Research Articles

Propagation of a cloud of hot electrons through a plasma in the presence of Langmuir scattering by ambient density fluctuations

Gas-dynamic theory is generalized to incorporate the effects of beam-driven Langmuir waves scattering off ambient density fluctuations, and the consequent effects on the propagation of a cloud of hot electrons in an inhomogeneous plasma. Assuming Langmuir scattering as the limit of nonlinear three-wave interactions with fluctuations that are weak, low-frequency, long-wavelength ion-sound waves, the net effect of scattering is equivalent to effective damping of the Langmuir waves. Under the assumption of self-similarity in the evolution of the beam and Langmuir wave distribution functions, gas-dynamic theory shows that the effects of Langmuir scattering on the beam distribution are equivalent to a perturbation in the injection profile of the beam. Analytical expressions are obtained for the height of the plateau of the beam distribution function, wave spectral number density, total wave and particle energy density, and the beam number density. The main results of gas-dynamic theory are then compared with simulation results from numerical solutions of quasilinear equations. The relaxation of the beam in velocity space is retarded in the presence of density fluctuations and the magnitude of the upper velocity boundary is less than that in the absence of fluctuations. There are four different regimes for the height of the plateau, corresponding to different stages of relaxation of the beam in velocity space. Moreover, Langmuir scattering results in transfer of electrons from moderate velocity to low velocity; this effect produces an enhancement in the beam number density at small distances near the injection site and a corresponding decrease at large distances. There are sharp decreases in the profiles of the beam and total wave energy densities, which are related to dissipation of energy at large phase velocities. Due to a slower velocity space diffusion of the beam distribution in the presence of scattering effects, the spatial width of the beam is reduced while its mean velocity of propagation increases slightly.

Inelastic scattering and interactions of three-wave parametric solitons

We study the interactions of velocity-locked three-wave parametric solitons in a medium with quadratic nonlinearity and dispersion. We reveal that the inelastic scattering between three-wave solitons and linear waves may be described in terms of analytical solutions with dynamically varying group velocity, or boomerons. Moreover, we demonstrate the elastic nature of three-wave soliton-soliton collisions and interactions.

Internal waves caused by the interaction of three harmonics under no restoring forces

The physical system under consideration consists of two fluids in uniform horizontal laminar motion parallel to their interface. All external restoring forces are absent. We find that nevertheless, three-wave interaction between different harmonics of the motion can take place. Attention is focused on the situation when this resonance is among the lowest three harmonics. Using the method of strained coordinates, a system of three coupled nonlinear partial differential equations is derived. These model the propagation of the interface, correct to second order. Solutions are found whose stability is briefly considered.

Nonlinear wave processes in a plasma with stationary random irregularities. Part I

We analyze the influence of large-scale random irregularities of a medium on nonlinear wave interactions. First, we study three-wave interactions in an equilibrium medium, and then we consider the problems of stabilizing the explosive instability in the presence of random irregularities.

Extended spectral scaling laws for shear-Alfvén wave turbulence

In this paper, resonant interactions among three shear-Alfvén waves is investigated in weak incompressible magnetohydrodynamics (MHD) turbulence, neglecting interactions involving the pseudo-Alfvén waves. It is found that these interactions lead to a wave kinetic equation that has the same compact form as the full nonlinear MHD system in the highly anisotropic limit (k⊥⪢k‖). These results have important implications for interpreting the effects of resonant three-wave interactions among shear-Alfvén waves throughout k space. In particular, it shows that resonant three-wave interactions involving only shear-Alfvén waves attempt to set up a k⊥nsf(k‖) 3D shear-Alfvén-wave spectrum (with −4<ns<−2, and f an arbitrary function depending on the initial conditions). The results also provide a framework for estimating the time scale on which the three shear-Alfvén-wave interactions transfer energy to larger k⊥ everywhere in the k⊥-k‖ plane. The compatibility of the results with previous predictions and the possible astrophysical implications of these extended spectral solutions for anisotropic plasmas is discussed.

Conservation style of the extra invariant for Rossby waves

We consider the dynamics of a system of Rossby waves with nonlinear interaction. It has been shown that such a system–besides the energy and momentum (enstrophy)–has an extra invariant, which is conserved approximately in the limit of small wave amplitudes. This invariant implies the anisotropy of the inverse cascade, when the energy is transferred from small scale eddies to the large scale zonal jets. The invariant has a quadratic (in wave amplitudes) term, which has a universal form, and a cubic term, whose form depends on the form of nonlinear interaction between Rossby waves. In this paper, we show that it is impossible to extend the extra invariant by higher order nonlinear terms (fourth order and higher) to obtain an exact conservation law. This fact holds irrespective of the form of nonlinearity (including the three-wave interaction). The extra invariant is similar to the adiabatic invariants in the theory of dynamical systems. We also show that on a “long time scale”, the cubic part of the invariant can be dropped (without sacrificing the accuracy of conservation), so that the extra invariant is given by the universal quadratic part. Finally, we show that the extra invariant mostly represents large scale modes, even to a greater extent than the energy. The presence of a small dissipation destroys the conservation of the enstrophy, which is based on small scale modes. At the same time, the extra invariant, along with the energy, remains almost conserved.

Nonlinear interaction of waves in the periodic structure with plasma-like layers

The three-wave interaction in an infinite periodic structure formed by two alternating semiconductor layers has been studied. In the present study, we analyze the peculiarities of the nonlinear interaction and their dependence on the parameters of periodic structure. The solution of the system of abridged equations is obtained. The laws of synchronism are studied. Conditions for the resonant interaction between the waves are analyzed.

A reduction of the resonant three-wave interaction to the generic sixth Painlevé equation

Among the reductions of the resonant three-wave interaction system to six-dimensional differential systems, one of them has been specifically mentioned as being linked to the generic sixth Painleve' equation P6. We derive this link explicitly, and we establish the connection to a three-degree of freedom Hamiltonian previously considered for P6.

Stable Control of Pulse Speed in Parametric Three-Wave Solitons

We analyze the control of the propagation speed of three wave packets interacting in a medium with quadratic nonlinearity and dispersion. We find analytical expressions for mutually trapped pulses with a common velocity in the form of a three-parameter family of solutions of the three-wave resonant interaction. The stability of these novel parametric solitons is simply related to the value of their common group velocity.

Three-Wave Interactions between Fast-Ion Modes in the National Spherical Torus Experiment

Simultaneous bursts of energetic particle mode (EPM) and toroidicity-induced Alfvén eigenmode (TAE) activity that correlate with significant fast-ion loss are observed in beam heated plasmas. Three-wave interactions between these modes are conclusively identified, indicating fixed phase relationships. This nonlinear coupling concentrates the energy of the TAEs into a toroidally localized perturbation frozen in the frame of a rigid, toroidally rotating structure formed by the EPMs. This redistribution of energy is significant because it will modify the effect of the TAEs on fast-ion loss.

All-optical signal processing using /spl chi//sup (2)/ nonlinearities in guided-wave devices

The authors present a review of all-optical signal-processing technologies based on /spl chi//sup (2)/ nonlinear interactions in guided-wave devices and their applications for telecommunication. In this study, the main focus is on three-wave interactions in annealed proton-exchanged periodically poled lithium niobate waveguides due to their suitable properties with respect to nonlinear mixing efficiency, propagation loss, and ease of fabrication. These devices allow the implementation of advanced all-optical signal-processing functions for next-generation networks with signal bandwidths beyond 1 THz. In this paper, integrated structures that will allow for improvements of current signal-processing functions as well as the implementation of novel device concepts are also presented.

Wave turbulence and vortices in Bose–Einstein condensation

We report a numerical study of turbulence and Bose–Einstein condensation within the two-dimensional Gross–Pitaevsky model with repulsive interaction. In the presence of weak forcing localized around some wave number in the Fourier space, we observe three qualitatively different evolution stages. At the initial stage a thermodynamic energy equipartition spectrum forms at both smaller and larger scales with respect to the forcing scale. This agrees with predictions of the four-wave kinetic equation of the Wave Turbulence (WT) theory. At the second stage, WT breaks down at large scales and the interactions become strongly nonlinear. Here, we observe formation of a gas of quantum vortices whose number decreases due to an annihilation process helped by the acoustic component. This process leads to formation of a coherent-phase Bose–Einstein condensate. After such a coherent-phase condensate forms, evolution enters a third stage characterised by three-wave interactions of acoustic waves that can be described again using the WT theory.

Simulation of three-wave interactions for focused beams with an assistant curvilinear coordinate system

We present a numerical model to deal with optical nonlinear processes at a strong focusing situation where general methods possibly fail to work. We find a special curvilinear coordinate system through which a new paraxial wave equation set is developed and its eigenmode transmission solutions are derived. On the basis of the wave equations, a detailed algorithm for second-harmonic generation is proposed with the Fourier-space method. Numerical results in non-walk-off and walk-off cases show that the new model can handle wave interactions on the strong focusing L/b>>1 condition within a relative small sampling grid, with which traditional methods will lose efficiency, and can provide unique features in comparison with the model in a Cartesian coordinate system. Concrete assessment of the new model is rendered by error analysis.

Wave turbulence in incompressible Hall magnetohydrodynamics

We investigate the steepening of the magnetic fluctuation power law spectra observed in the inner solar wind for frequencies higher than 0.5 Hz. This high frequency part of the spectrum may be attributed to dispersive nonlinear processes. In that context, the long-time behavior of weakly interacting waves is examined in the framework of 3D incompressible Hall MHD turbulence. The Hall term added to the standard MHD equations makes the Alfv\'en waves dispersive and circularly polarized. We introduce the generalized Els\"asser variables and, using a complex helicity decomposition, we derive for three-wave interaction processes the general wave kinetic equations; they describe the nonlinear dynamics of Alfv\'en, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field $B_0 \ep$. Hall MHD turbulence is characterized by anisotropies of different strength. We show that electron and standard MHD turbulence can be seen as two frequency limits of the present theory but the standard MHD limit is singular; additionally, we analyze in detail the ion MHD turbulence limit. Exact power law solutions of the master wave kinetic equations are given in the small and large scale limits for which we have, respectively, the total energy spectra $E(\kpn,\kpa) \sim \kpn^{-5/2} |\kpa|^{-1/2}$ and $E(\kpn,\kpa) \sim \kpn^{-2}$. An anisotropic phenomenology is developed to describe continuously the different scaling laws of the energy spectrum; one predicts $E(\kpn,\kpa) \sim \kpn^{-2} |\kpa|^{-1/2} (1+\kpn^2d_i^2)^{-1/4}$. Nonlocal interactions between Alfv\'en, whistler and ion cyclotron waves are investigated; a non trivial dynamics exists only when a discrepancy from the equipartition between the large scale kinetic and magnetic energies happens.

Holographic light scattering in centrosymmetric sodium nitroprusside upon generation of light-induced metastable states

Received 16 December 2005; published 22 May 2006The phenomenon of holographic light scattering in centrosymmetric crystals of sodiumnitroprusside isinvestigated by analyzing 1 the effect of light polarization and wavelength of pump- and readout beams, 2the angular intensity distribution, and 3 the kinetics of scattered light. It is shown that the scattering phe-nomenon and the corresponding properties can be entirely explained by diffraction processes of the pump beamfrom parasitically recorded slanted phase gratings without assuming any beam-coupling processes. The kineticsoriginates from the nonlinear photorefractive recording mechanism. Further we present an approach for three-wave interaction including diffraction processes at higher harmonics, that explains the temporal build-up ofwide-angle scattering in media with a nonlinear photorefractive response.DOI: 10.1103/PhysRevB.73.205113 PACS number s : 42.40. i, 42.65.Pc, 42.70.Ln, 78.35. c

Grid states and nonlinear selection in parametrically excited surface waves

Interacting surface waves, parametrically excited by two commensurate frequencies (Faraday waves), yield a rich family of nonlinear states, which result from a variety of three-wave resonant interactions. By perturbing the system with a third frequency, we selectively favor different nonlinear wave interactions. Where quadratic nonlinearities are dominant, the only observed patterns correspond to "grid" states. Grid states are superlattices in which two corotated sets of critical wave vectors are spanned by a sublattice whose basis states are linearly stable modes. Specific driving phase combinations govern the selection of different grid states.

Enhanced Faraday pattern stability with three-frequency driving

We report experimental observations of enhanced stability of quasipatterns and superlattice patterns in a vertically oscillated, deep viscous fluid layer with the addition of a third driving frequency. With two-frequency driving in the ratios 4:5 and 6:7, 12-fold quasipatterns and type-I superlattice patterns appear, respectively, as a secondary instability for a range of relative phases and amplitudes. Addition of a small third-frequency component at twice the difference frequency, i.e., 4:5:2 and 6:7:2, shifts the region of stability for these patterns closer to onset. For a range of parameter values the stabilized patterns become the primary instability. The degree of stabilization is sensitive to the amplitude and relative phase of the third-frequency term in qualitative agreement with a recent symmetry based analysis of resonant three-wave interactions.

Anisotropic weak turbulence of Alfvén waves in collisionless astrophysical plasmas

The evolution of Alfven turbulence due to three-wave interactions is discussed using kinetic theory for a collisionless, thermal plasma. There are three low-frequency modes, analogous to the three modes of compressible MHD. When only Alfven waves are considered, the known anisotropy of turbulence in incompressible MHD theory is reproduced. Inclusion of a fast mode wave leads to separation of turbulence into two regimes: small wave numbers where three-wave processes involving a fast mode is dominant, and large wave numbers where the three Alfven wave process is dominant. Possible application of the anisotropic Alfven turbulence to the interstellar medium and dissipation of magnetic energy in magnetars is discussed.

Nonlinear three-mode interaction and drift-wave turbulence in a tokamak edge plasma

A three-wave interaction model with quadratic nonlinearities and linear growth/decay rates is used to investigate the occurrence of drift-wave turbulence driven by pressure gradients in the edge plasma of a tokamak. Model parameters are taken from a typical set of measurements of the floating electrostatic potential in the tokamak edge region. Some aspects of the temporal dynamics exhibited by the three-wave interaction model are investigated, with special emphasis on a chaotic regime found for a wide range of the wave decay rate. An intermittent transition from periodic to chaotic behavior is observed and some statistical properties, such as the interburst and laminar length interval durations, are explored.

Nonlinear evolution of components of an electromagnetic field of helicoidal waves in plasma

We derive model equations of the interaction between circular polarized helicon wavetrains in a two-dimensional plasma. For such quasi-one-dimensional waves a nonlinear resonance results in N-wave interactions. The corresponding system of equations is known to be integrable. A novel feature of the study, in the context of soliton theory, is the account of the wave asynchronism that seems to be essential for wavetrains with wavefronts of large steepness, observed in experiments for dense helicon plasma. To generate solutions we use recent results of the binary Darboux transformation technique, applying it to three-wave interactions. The solutions have been analysed numerically and plotted.