Three-wave Resonant Interaction Research Articles

Formation of Rogue Waves and Modulational Instability with Zero-Wavenumber Gain in Multicomponent Systems with Coherent Coupling.

It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Starting with the Bers-Kaup-Reiman system for three-wave resonant interactions, we identify a specific RW-building mechanism based on MI which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by MI in other multicomponent systems-in particular, in the massive Thirring model.

Non-Linear Interaction of Harmonic Waves in a Quasi-Isentropic Flow of Magnetic Gas

The diversity of wave modes in the magnetic gas gives rise to a wide variety of nonlinear phenomena associated with these modes. We focus on the planar fast and slow magnetosound waves in the geometry of a flow where the wave vector forms an arbitrary angle θ with the equilibrium straight magnetic field. Nonlinear distortions of a modulated signal in the magnetic gas are considered and compared to that in unmagnetised gas. The case of acoustical activity of a plasma is included into consideration. The resonant three-wave non-collinear interactions are also discussed. The results depend on the degree of non-adiabaticity of a flow, θ, and plasma-β.

Binary Darboux transformation, solitons and breathers for a second-order three-wave resonant interaction system

Binary Darboux transformation, solitons and breathers for a second-order three-wave resonant interaction system

Fermi-Pasta-Ulam-Tsingou recurrence and cascading mechanism for resonant three-wave interactions.

Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. Here we study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT). While nonlinearity governs the actual dynamics, the range of wave numbers for modulation instability remains a useful indicator. Depending on the stability characteristics of the fundamental mode and the higher-order harmonics ("sidebands"), "regular" and "staggered" FPUT patterns can arise. A "cascading mechanism" provides analytical verification, as the fundamental and sideband modes attain the same magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also computed, similar to experimental observations of optical pulses. Such spectra can elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of "effective energy" is examined and the eigenvalues of the inverse scattering mechanism are computed. Both approaches are utilized to correlate with the occurrence of regular or staggered FPUT. These numerical and analytical studies can enhance our understanding of wave interactions in fluid mechanics and optics.

Evidence of experimental three-wave resonant interactions between two dispersion branches.

We report the observation of nonlinear three-wave resonant interactions between two different branches of the dispersion relation of hydrodynamic waves, namely the gravity-capillary and sloshing modes. These atypical interactions are investigated within a torus of fluid for which the sloshing mode can be easily excited. A triadic resonance instability is then observed due to this three-wave two-branch interaction mechanism. An exponential growth of the instability and phase locking are evidenced. The efficiency of this interaction is found to be maximal when the gravity-capillary phase velocity matches the group velocity of the sloshing mode. For a stronger forcing, additional waves are generated by a cascade of three-wave interactions populating the wave spectrum. Such a three-wave two-branch interaction mechanism is probably not restricted to hydrodynamics and could be of interest in other systems involving several propagation modes.

Rogue waves of wave resonant-interaction equations with the higher-order effects

We study a three coupled wave system which includes the vector Hirota equations and three-wave resonant interaction equations. Using the Darboux-Dressing transformation, we obtain the rational and semi-rational (mixed rational-exponential) vector solutions of such system on the finite backgrounds and mixed backgrounds for the purpose of modeling rogue waves. Besides, we show the dynamics of vector solitons to this three coupled wave system.

Non-degenerate rogue waves and multiple transitions in systems of three-wave resonant interaction

Non-degenerate rogue waves (RWs) and multiple transitions between RWs and solitons arising from vector three-wave resonant interaction are studied analytically and numerically. In contrast to the conventional degenerate RWs, such non-degenerate RWs consist of two fundamental RWs each with individual Lax spectral parameter. We show distinctive continuous transitions from bright (dark) RWs to four-petal RWs to dark (bright) solitons as the relative frequency between the wave components increases. The underlying mechanism of such processes is the non-monotonic variation of the energy exchanges between different components of the waves. We further reveal the transition dynamics of non-degenerate RWs. Finally, we confirm numerically the robustness of the transition dynamics in the presence of spontaneous modulation instability induced by white noise. Our results provide insights into the RW formation and the multiple transitions in systems with three-wave resonant interaction, and may offer the possibility of experimental observations in multi-component resonant processes.

Formula omitted]-fold generalized Darboux transformation and soliton interactions for a three-wave resonant interaction system in a weakly nonlinear dispersive medium

In this paper, a three-wave resonant interaction system, which describes the resonant mixing of the waves in a weakly nonlinear dispersive medium, is studied. Starting from the first-order Darboux transformation, we construct an N-fold generalized Darboux transformation (GDT) in which n different spectral parameters are involved, where n and N are the positive integers, and n≤N. Utilizing the obtained N-fold GDT, we derive three types of the Y-shaped bright-dark-bright solitons. Those solitons have different characteristic lines as follows: three rays; one ray and two curves; one ray, one line and two curves. Interactions among the three kinds of Y-shaped bright-dark-bright solitons are graphically illustrated. Those interactions are shown to be elastic.

Novel soliton solutions for the fractional three-wave resonant interaction equations

Abstract In this article, we obtained new infinite sets of exact soliton solutions for the nonlinear evolution system of three-wave resonant interaction equations. The solved system contains the non-zero second-order dispersion coefficients, the non-zero phase velocity mismatch, and the conformable fractional time derivative of order between zero and one. The solution method is a constructed ansatz that consists of linear combinations of the tan and cotan hyperbolic functions with complex coefficients. We stated clear systematic steps toward writing an exact soliton solution for the studied system. To show the efficiency of this method, we introduced some numerical examples on each obtained set of solutions. The computations showed that similar solutions can be obtained if one replaces the tan and cotan hyperbolic functions with the tan and cotan trigonometric functions. The new obtained fractional solutions could be useful in studying the broad applications of triad resonances in plasma physics and in nonlinear optics.

Explosive instability of dust settling in a protoplanetary disc

ABSTRACT It is shown that gas-dust perturbations in a disc with dust settling to the disc mid-plane exhibit the non-linear three-wave resonant interactions between streaming dust wave (SDW) and two inertial waves (IW). In the particular case considered in this paper, SDW at the wavenumber k• = 2κ/(g$z$ts), where κ, g$z$, and ts are, respectively, epicyclic frequency, vertical gravitational acceleration, and particle’s stopping time, interacts with two IW at the lower wavenumbers k′ and k″ such that k′ < kDSI < k″ < k•, where kDSI = κ/(g$z$ts) is the wavenumber of the linear resonance between SDW and IW associated with the previously discovered linear dust settling instability. The problem is solved analytically in the limit of the small dust fraction. As soon as the dynamical dust back reaction on gas is taken into account, k•, k′, and k″ become slightly non-collinear and the emerging interaction of waves leads to simultaneous explosive growth of their amplitudes. This growth is explained by the conservative exchange with energy between the waves. The amplitudes of all three waves grow because the negative energy SDW transfers its energy to the positive energy IW. The product of the dimension-less amplitude of initially dominant wave and the time of explosion can be less than Keplerian time in a disc. It is shown that, generally, the three-wave resonance of an explosive type exists in a wide range of wavenumbers 0 < k• ≤ 2κ/(g$z$ts). An explosive instability of gas-dust mixture may facilitate the dust clumping and the subsequent formation of planetesimals in young protoplanetary discs.

Rogue waves in (2+1)-dimensional three-wave resonant interactions

Rogue waves in (2+1)-dimensional three-wave resonant interactions are studied by the bilinear KP-reduction method. General rogue waves arising from a constant background, from a lump-soliton background and from a dark-soliton background have been derived, and their dynamics illustrated. For rogue waves arising from a constant background, fundamental rogue waves are line-shaped, and multi-rogue waves exhibit multiple intersecting lines. Higher-order rogue waves could also be line-shaped, but they exhibit multiple parallel lines. For rogue waves arising from a lump-soliton background, they could exhibit distinctive patterns due to their interaction with the lump soliton. For rogue waves arising from a dark-soliton background, their intensity pattern could feature half-line shapes or lump shapes, which are very novel.

N-Fold Generalized Darboux Transformation and Soliton Interactions for a Three-Wave Resonant Interaction System in a Weakly Nonlinear Dispersive Medium

N-Fold Generalized Darboux Transformation and Soliton Interactions for a Three-Wave Resonant Interaction System in a Weakly Nonlinear Dispersive Medium

N-Fold Generalized Darboux Transformation and Soliton Interactions for a Three-Wave Resonant Interaction System in a Weakly Nonlinear Dispersive Medium

N-Fold Generalized Darboux Transformation and Soliton Interactions for a Three-Wave Resonant Interaction System in a Weakly Nonlinear Dispersive Medium

Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system

<abstract><p>The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.</p></abstract>

Three-wave resonant interactions: dark–bright–bright mixed N- and high-order solitons, breathers, and their structures

In this paper, a three-wave resonant interaction system, which describes the mixture of waves with different frequencies in a weakly nonlinear and dispersive medium, is investigated with the help of the Darboux transformation and its extension. Firstly, we construct the dark–bright–bright mixed multi-soliton solutions in terms of the determinant. Dark–bright–bright solitons are found to arise in the focusing, defocusing and mixed cases. Bound state of the two dark–bright–bright solitons is also analyzed. Secondly, we derive the dark–bright–bright mixed high-order semi-rational solitons and find that their structures are different from the multi-soliton ones. Finally, we present the Nth-order breather solutions in terms of the determinant, where N is a positive integer. Line-shaped and three wing-shaped structures of the breathers are obtained.

An alternative form of the fundamental plasma emission through the coalescence of Z-mode waves with whistlers

Plasma emission (PE), i.e., electromagnetic radiation at the plasma frequency and its second harmonic, is a general process occurring in both astrophysical and laboratory plasmas. The prevailing theory presents a multi-stage process attributed to the resonant coupling of beam-excited Langmuir waves with ion-acoustic waves. Here, we examine another possibility of the fundamental PE induced by the resonant coupling of Z-mode and whistler (W) waves. Earlier studies have been controversial in the plausibility and significance of such process in plasmas. In this study, we show that the matching condition of three-wave resonant interaction (Z + W → O) can be satisfied over a wide regime of parameters based on the magnetoionic theory, demonstrate the occurrence of such process, and further evaluate the rate of energy conversion from the pumped Z or W mode to the fundamental O mode with particle-in-cell simulations of wave pumping. The study presents an alternative form of the fundamental PE, which could possibly play a role in various astrophysical and laboratory scenarios with both Z and W modes readily excited through the electron cyclotron maser instability.

The Riemann–Hilbert approach to the generalized second-order flow of three-wave hierarchy

This paper extends the second-order three-wave flow of Fordy and Kulish to propose a generalized second-order flow of the three-wave hierarchy. Then the Lax pair of the generalized second-order three-wave flow is constructed by the standard prolongation technique, and the infinite conservation laws for this flow are derived. Finally, the Riemann–Hilbert approach is applied to the initial value problem of this flow, and the N-soliton solutions to the second-order three-wave flow are derived explicitly. Collision behaviors of the two and three solitons are demonstrated graphically, which shows that our results have potential applications to the resonant three-wave interactions in nonlinear media.

General rogue waves in the three-wave resonant interaction systems

Abstract General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction, respectively. It is shown that while the first family of solutions associated with a simple root exists for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, the ones generated by a $2\times 2$ block determinant in the double-root case, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.

On the Role of the Explosive Interaction of Three Surface Waves at the Initial Stage of Spray Generation in Strong Winds

This work is devoted to a theoretical study of the hydrodynamic instability of the water–air interface. The development of this instability may result in “bag-breakup” fragmentation, which is one of the main sources of droplets in hurricane winds. A hypothesis according to which the initial water surface elevations subjected to fragmentation are formed by the hydrodynamic instability of disturbances of the wind drift current in the water is proposed. The weakly nonlinear stage of instability in the form of a resonant three-wave interaction is considered. It is found that the nonlinear resonant interaction of a triad of wind drift perturbations, one wave of which is directed along the flow and two other waves are directed at an angle to the flow, leads to an explosive increase in amplitudes. As part of the piecewise continuous model of the drift current profile, the characteristic time and spatial scales of disturbances are found and it is shown that their characteristic dependences on the air friction velocity agree with previously obtained experimental data.

Plural Three-Wave Resonant Interactions in the Transit Section of Two-Stream Superheterodyne FEL with a Longitudinal Electric Field

Plural Three-Wave Resonant Interactions in the Transit Section of Two-Stream Superheterodyne FEL with a Longitudinal Electric Field