Three-wave Resonant Interaction Research Articles

Appearance of an ensemble solitary pulse train in transient backward stimulated Raman scattering

A self-similarity approximation model for three-wave resonant interaction in transient backward stimulated Raman scattering (BSRS) was derived. Interestingly, as its asymptotic solution, the train of solitary backward Raman pulse train is possibly formed. This result can be considered as an increment of “single” soliton observed previously [A. Abdolvand et al., Phys. Rev. Lett 103, 183902 (2009)].

Growth and damping of interfacial waves on a diffuse interface

The resonant interaction of surface and internal waves produces a nonlinear mechanism for energy transfer among wave components in oceans, lakes, and estuaries. In many field situations, the stratification may be well approximated by a two-layer fluid with a diffuse interface. The growth and damping rates of sub-harmonic interfacial waves generated by a surface wave through a three-wave resonant interaction are measured in the laboratory. These measurements are compared with theoretical predictions. A diffuse interface reduces the damping rate and increases the growth rate. The predicted growth rate provides excellent comparison with the laboratory measurements. The inclusion of the effects of a diffuse interface significantly improve the comparison.

Optical rogue waves in parametric three-wave mixing and coherent stimulated scattering

We provide a systematic study of the rogue-wave dynamics in phase-matched weakly dispersive nonlinear media, which arise either from the parametric mixing or from the stimulated backscattering process, both of which are modeled by the three-wave resonant interaction equations. The explicit rogue-wave solutions up to the second order are presented. These solutions reveal the markedly different vector rogue-wave behaviors of the parametric and the scattering processes. In the degenerate situation where two interacting optical fields share the same group velocity, the sum of the intensities of this pair of optical wave components is spatiotemporally invariant. The close link between the existence of rogue-wave solutions and the presence of baseband modulation instability is confirmed. Finally, numerical simulations confirm that the generation of vector rogue waves is robust in the presence of spontaneous modulation instability activated by quantum noise.

Higher-order rogue wave solutions of the three-wave resonant interaction equation via the generalized Darboux transformation

In this paper, we utilize generalized Darboux transformation to study higher-order rogue wave solutions of the three-wave resonant interaction equation, which describes the propagation and mixing of waves with different frequencies in weakly nonlinear dispersive media. A general Nth-order rogue wave solution with two characteristic velocities structural parameters and 3N independent parameters under a determined plane-wave background and a specific parameter condition is derived. As an application, we show that four fundamental rogue waves with fundamental, two kinds of line and quadrilateral patterns, or six fundamental rogue waves with fundamental, triangular, two kinds of quadrilateral and circular patterns can emerge in the second-order rogue waves. Moreover, several important wave characteristics including the maximum values, the corresponding coordinate positions of the humps, and the stability problem for some special higher-order rogue wave solutions such as the fundamental and quadrilateral cases are discussed.

Novel systems of resonant wave interactions

A matrix Riemann–Hilbert problem (RHP) is constructed using the dressing method starting from two uncoupled, one-directional linear wave equations; the RHP thus obtained is then used to derive a novel integrable matrix non-local system of equations describing resonant wave interactions, together with its Lax pair. This system is shown to be a matrix generalization of the equations for resonant three-wave interactions and stimulated Raman scattering. Several compatible reductions admitted by this system are also discussed.

Nonlocal resonances in weak turbulence of gravity-capillary waves.

We report a laboratory investigation of weak turbulence of water surface waves in the gravity-capillary crossover. By using time-space-resolved profilometry and a bicoherence analysis, we observe that the nonlinear processes involve three-wave resonant interactions. By studying the solutions of the resonance conditions, we show that the nonlinear interaction is dominantly one dimensional and involves collinear wave vectors. Furthermore, taking into account the spectral widening due to weak nonlinearity explains why nonlocal interactions are possible between a gravity wave and high-frequency capillary ones. We observe also that nonlinear three-wave coupling is possible among gravity waves, and we raise the question of the relevance of this mechanism for oceanic waves.

Weak magnetohydrodynamic turbulence and intermittency

Three-dimensional numerical simulation is used to investigate intermittency in incompressible weak magnetohydrodynamic turbulence with a strong uniform magnetic field $\boldsymbol{b}_{\mathbf{0}}$ and zero cross-helicity. At leading order, this asymptotic regime is achieved via three-wave resonant interactions with the scattering of a wave on a 2D mode for which $k_{\Vert }=0$. When the interactions with the 2D modes are artificially reduced, we show numerically that the system exhibits an energy spectrum with $k_{\bot }^{-3/2}$, whereas the expected exact solution with $k_{\bot }^{-2}$ is recovered with the full nonlinear system. In the latter case, strong intermittency is found when the vector separation of structure functions is taken transverse to $\boldsymbol{b}_{\mathbf{0}}$. This result may be explained by the influence of the 2D modes whose regime belongs to strong turbulence. In addition to shedding light on the origin of this intermittency, we derive a log-Poisson law, ${\it\zeta}_{p}=p/8+1-(1/4)^{p/2}$, which fits the data perfectly and highlights the important role of parallel current sheets.

Multiscale asymptotics for the Skeleton of the Madden-Julian Oscillation and Tropical–Extratropical Interactions

AbstractAnew model is derived and analyzed for tropical–extratropical interactions involving the Madden– Julian oscillation (MJO). The model combines (i) the tropical dynamics of the MJO and equatorial baroclinic waves and (ii) the dynamics of barotropic Rossby waves with significant extratropical structure, and the combined system has a conserved energy. The method of multiscale asymptotics is applied to systematically derive a system of ordinary differential equations (ODEs) for three-wave resonant interactions. Two novel features are (i) a degenerate auxiliary problem with overdetermined equations due to a compatibility condition (meridional geostrophic balance) and (ii) cubic self-interaction terms that are not typically found in threewave resonance ODEs. Several examples illustrate applications to MJO initiation and termination, including cases of (i) the MJO, equatorial baroclinic Rossbywaves, and barotropic Rossbywaves interacting, and (ii) the MJO, baroclinic Kelvinwaves, and barotropic Rossbywaves interacting. Resonance with the Kelvinwave is not possible here if only dry variables are considered, but it occurs in the moist model here through interactions with water vapor and convective activity.

On the formation of structures in nonequilibrium media in the resonant three-wave interaction

A mechanism of formation of flat and spiral solitary structures in nonequilibrium media allowing the resonant three-wave interaction of eigenmodes is proposed. A bifurcations diagram on the 'pump wave amplitude - initial profile width' plane is plotted

Nonlocal nonlinear coupling of kinetic sound waves

Abstract. We study three-wave resonant interactions among kinetic-scale oblique sound waves in the low-frequency range below the ion cyclotron frequency. The nonlinear eigenmode equation is derived in the framework of a two-fluid plasma model. Because of dispersive modifications at small wavelengths perpendicular to the background magnetic field, these waves become a decay-type mode. We found two decay channels, one into co-propagating product waves (forward decay), and another into counter-propagating product waves (reverse decay). All wavenumbers in the forward decay are similar and hence this decay is local in wavenumber space. On the contrary, the reverse decay generates waves with wavenumbers that are much larger than in the original pump waves and is therefore intrinsically nonlocal. In general, the reverse decay is significantly faster than the forward one, suggesting a nonlocal spectral transport induced by oblique sound waves. Even with low-amplitude sound waves the nonlinear interaction rate is larger than the collisionless dissipation rate. Possible applications regarding acoustic waves observed in the solar corona, solar wind, and topside ionosphere are briefly discussed.

Algebro-geometric Quasi-periodic Solutions to the Three-Wave Resonant Interaction Hierarchy

Based on two sets of the Lenard recursion gradients, we derive the three-wave resonant interaction (TWRI) hierarchy associated with a $3\times 3$ matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the TWRI hierarchy, we introduce a trigonal curve ${\cal K}_{m-2}$ of arithmetic genus $m-2$ with three infinite points and establish the corresponding Baker--Akhiezer function and meromorphic functions on ${\cal K}_{m-2}$. The TWRI equations are decomposed into a system of Dubrovin-type ordinary differential equations. Using the theory of the trigonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit theta function representations of the Baker--Akhiezer function, of the meromorphic functions, and, in particular, of solutions for the entire TWRI hierarchy.

Soliton decay in composite right- and left-handed transmission lines periodically loaded with Schottky varactors

Soliton decay in a composite rightand left-handed transmission line periodically loaded with Schottky varactors, termed a Schottky CRLH line, is characterized to obtain broadband envelope pulses. When the three waves in the line satisfy the wave-number and frequency resonance conditions, they are governed by the three-wave resonant interaction equation, which have solutions that exhibit soliton decay, i.e., the decay of the highest frequency envelope into those carried by the other two frequency solitons. Due to the left-handedness, solitons resulting from the decay can have much smaller width than the incident envelope. This paper discusses the potential of a Schottky CRLH line as a generator of broadband envelope pulses.

Magnetorotational Decay Instability in Keplerian Disks

The saturation of the magnetorotational instability (MRI) in thin Keplerian disks through three-wave resonant interactions is introduced and discussed. That mechanism is a natural generalization of the fundamental decay instability discovered five decades ago for infinite, homogeneous, and immovable plasmas. The decay instability relies on the energy transfer from the MRI to stable slow Alfvén-Coriolis as well as magnetosonic waves. A second-order forced Duffing amplitude equation for the initially unstable MRI as well as two first-order equations for the other two waves are derived. The solutions of those equations exhibit bounded bursty nonlinear oscillations for the MRI as well as unbounded growth for the linearly stable slow Alfvén-Coriolis and magnetosonic perturbations, thus giving rise to the magnetorotational decay instability.

Rogue Waves Emerging from the Resonant Interaction of Three Waves

We introduce a novel family of analytic solutions of the three-wave resonant interaction equations for the purpose of modeling unique events, i.e., "amplitude peaks" which are isolated in space and time. The description of these solutions is likely to be a crucial step in the understanding and forecasting of rogue waves in a variety of multicomponent wave dynamics, from oceanography to optics and from plasma physics to acoustics.

Complete energy conversion by autoresonant three-wave mixing in nonuniform media

Resonant three-wave interactions appear in many fields of physics e.g. nonlinear optics, plasma physics, acoustics and hydrodynamics. A general theory of autoresonant three-wave mixing in a nonuniform media is derived analytically and demonstrated numerically. It is shown that due to the medium nonuniformity, a stable phase-locked evolution is automatically established. For a weak nonuniformity, the efficiency of the energy conversion between the interacting waves can reach almost 100%. One of the potential applications of our theory is the design of highly-efficient optical parametric amplifiers.

Evolution of sub-harmonic interfacial waves generated by a surface wave through a three-wave interaction

Higher-order effects in three-wave resonant interaction of a surface wave, with a pair of interfacial waves in a two-layer fluid, are studied theoretically. Following an initial rapid growth, the interfacial waves approach a steady state of constant amplitude. An explicit solution is presented for transition to the ultimate state of the interaction. It is shown that for interaction in a wave flume, it is necessary to include a 2nd pair of the interfacial waves, resulting from a reflection of the original pair in the analysis. The effects of different parameters on the dynamics of the interaction are investigated. The results indicate that a faster initial growth does not necessarily lead to larger ultimate amplitude. Also, there are two angles, at which the interfacial waves continue to grow at an initial growth rate, possibly leading to wave breaking. The results are in qualitative agreement with previous experimental observations.

On stochastic stabilization of the Kelvin-Helmholtz instability by three-wave resonant interaction

An analytical investigation of the effect of three-wave resonant interactions with the linearly unstable wave is proposed. We consider the waves in the Kelvin-Helmholtz model, consisting of two fluid layers with different densities and velocities. We suppose that the velocity shear is weakly supercritical, the instability is of the algebraic type, i.e., the amplitude of the unstable wave grows linearly, and the instability occurs within the framework of a single mode. The amplitudes of two other waves taking part in the nonlinear interaction are assumed to be stable. The initial amplitudes of these waves are supposed to be small in comparison with the initial amplitude of the unstable wave. We present an analysis of the system of amplitude equations derived for this case using JWKB-method. As a result, we obtain equations that couple solutions pre- and post-passing the singular point, i.e., the point where the amplitude of the unstable wave has a local minimum. These equations give us the transformation rule of a parameter that characterizes the phase shift between fast and slow waves and defines the behavior of the system. This parameter is constant between two singular points and varies by chance at a singular point. As long as it stays positive, the amplitude of the wave remains limited and performs stochastic oscillations. If this parameter passes over zero, then we leave the region of stabilization and turn out in the region, where the amplitude grows infinitely. Accordingly, the transition to the region of instability happens stochastically. However, if the time interval, when the amplitude remains bounded, is large enough, the proposed scenario can be treated as a partial stabilization of instability.

Experimental observation of standing interfacial waves induced by surface waves in muddy water

A striking feature has been observed in a laboratory wave tank with a thin layer of clear water overlying a layer of mud. A piston-type wave maker is used to generate long monochromatic surface waves in a tank with a layer of kaolinite clay at the bottom. The wave action on the mud causes the clay particles to rise from the bottom into the water column, forming a lutocline. As the lutocline approaches the water surface, a set of standing interfacial waves form on the lutocline. The interfacial wave directions are oriented nearly orthogonal to the surface wave direction. The interfacial waves, which sometimes cover the entire length and width of the tank, are also temporally subharmonic as the phase of the interfacial wave alternates with each passing surface wave crest. These interfacial waves are the result of a resonant three-wave interaction involving the surface wave train and the two interfacial wave trains. The interfacial waves are only present when the lutocline is about 3 cm of the water surface and they can be sufficiently nonlinear as to exhibit superharmonics and a breaking-type of instability.

Pattern formation on the surface of a bubble driven by an acoustic field

The final stable shape taken by a fluid–fluid interface when it experiences a growing instability can be important in determining features as diverse as weather patterns in the atmosphere and oceans, the growth of cell structures and viruses, and the dynamics of planets and stars. An example which is accessible to laboratory study is that of an air bubble driven by ultrasound when it becomes shape-unstable through a parametric instability. Above the critical driving pressure threshold for shape oscillations, which is minimal at the resonance of the breathing mode, regular patterns of surface waves are observed on the bubble wall. The existing theoretical models, which take account only of the interaction between the breathing and distortion modes, cannot explain the selection of the regular pattern on the bubble wall. This paper proposes an explanation which is based on the consideration of a three-wave resonant interaction between the distortion modes. Using a Hamiltonian approach to nonlinear bubble oscillation, corrections to the dynamical equations governing the evolution of the amplitudes of interacting surface modes have been derived. Steady-state solutions of these equations describe the formation of a regular structure. Our predictions are confirmed by images of patterns observed on the bubble wall.

Superheterodyne amplification of electromagnetic waves of optical and terahertz bands in gallium nitride films

Superheterodyne amplification of electromagnetic waves of optical and terahertz bands in the case of three-wave interaction in n-GaN films with the space change wave of millimeter band amplified due to negative differential resistance is studied. It is shown that amplification of the space change wave in n-GaN films may be achieved on higher frequencies f ≤ 500 GHz than when using GaAs. The case of three-wave resonant interaction of two counter-propagating waves with the space charge wave is considered for the waveguide on based on GaN film on dielectric substrate. It is shown that gain of electromagnetic waves of optical band may reach 20–40 dB on the waveguide lengths of up to 100 μm.