Three-wave Interaction System Research Articles

Second-harmonic generation in branched optical waveguides: Metric graphs based approach

We consider the second-harmonic generation in branched optical waveguides, by modeling the process in terms of the three-wave interaction system on metric graphs. The vertex boundary conditions are derived from the conservation laws. Soliton solutions of the three-wave interaction system are obtained for the case of star- and tree-branched networks.

A second-order three-wave interaction system and its rogue wave solutions

A second-order three-wave interaction system and its rogue wave solutions

Amplification and Compression of Radiation Emitted by a Dense Relativistic Cold Electron Beam

Starting from a three-wave interaction system of equations for free-electron lasers in the framework of a quantum fluid model, we show that these equations satisfy the Sine-Gordon equation. The full solution in space and in time of this set of equations are numerically obtained.

Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media.

We study the three-wave interaction that couples an electromagnetic pump wave to two frequency down-converted daughter waves in a quadratic optical crystal and PT-symmetric potentials. PT symmetric potentials are shown to modulate stably nonlinear modes in two kinds of three-wave interaction models. The first one is a spatially extended three-wave interaction system with odd gain-and-loss distribution in the channel. Modulated by the PT-symmetric single-well or multi-well Scarf-II potentials, the system is numerically shown to possess stable soliton solutions. Via adiabatical change of system parameters, numerical simulations for the excitation and evolution of nonlinear modes are also performed. The second one is a combination of PT-symmetric models which are coupled via three-wave interactions. Families of nonlinear modes are found with some particular choices of parameters. Stable and unstable nonlinear modes are shown in distinct families by means of numerical simulations. These results will be useful to further investigate nonlinear modes in three-wave interaction models.

A theoretical study of transient stimulated Brillouin scattering in optical fibers seeded with phase-modulated light

Beam combining of phase-modulated kilowatt fiber amplifiers has generated considerable interest recently. We describe in the time domain how stimulated Brillouin scattering (SBS) is generated in an optical fiber under phase-modulated laser conditions, and we analyze different phase modulation techniques. The temporal and spatial evolutions of the acoustic phonon, laser, and Stokes fields are determined by solving the coupled three-wave interaction system. Numerical accuracy is verified through agreement with the analytical solution for the un-modulated case and through the standard photon conservation relation for counter-propagating optical fields. As a test for a modulated laser, a sinusoidal phase modulation is examined for a broad range of modulation amplitudes and frequencies. We show that, at high modulation frequencies, our simulations agree with the analytical results obtained from decomposing the optical power into its frequency components. At low modulation frequencies, there is a significant departure due to the appreciable cross talk among the laser and Stokes sidebands. We also examine SBS suppression for a white noise source and show significant departures for short fibers from analytically derived formulas. Finally, SBS suppression through the application of pseudo-random bit sequence modulation is examined for various patterns. It is shown that for a fiber length of 9 m the patterns at or near n=7 provide the best mitigation of SBS with suppression factors approaching 17 dB at a modulation frequency of 5 GHz.

Invariant algebraic surfaces for the reduced three-wave interaction system

In this paper, we consider the three-wave interaction system x=γx+δy+z−2y2,y=γy−δx+2xy,z=−2z−2zx, where γ, δ are real parameters. Our main results are the complete characterization of all values of the parameters γ, δ for which the three-wave interaction system admits either invariant algebraic surfaces or algebraic (polynomial or rational) first integrals.

Systematic method of generating new integrable systems via inverse Miura maps

We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1 + 1 dimensions as well as in 2 + 1 dimensions. The effectiveness of the method is illustrated using examples such as the nonlinear Schrödinger (NLS) system, the Zakharov–Ito system (two-component KdV), the three-wave interaction system, the Yajima–Oikawa system, the Ablowitz–Ladik lattice (integrable space-discrete NLS), and two (2 + 1)-dimensional NLS systems.

Integrals of motion of the reduced three-wave interaction system

By using the Darboux method of integrability and solving linear partial differential equations, the whole classification of the integrals of motion of the reduced three-wave interaction system is obtained. Rigorous proof is given.

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.

Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances

The nonlinear Schrödinger (NLS) equation can be derived as an amplitude equation describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. The purpose of this paper is to prove estimates, between the formal approximation, obtained via the NLS equation, and true solutions of the original system in case of non-trivial quadratic resonances. It turns out that the approximation property (APP) holds if the approximation is stable in the system for the three-wave interaction (TWI) associated to the resonance. We construct a counterexample showing that the NLS equation can fail to approximate the original system if instability occurs for the approximation in the TWI system. In the unstable case we give some arguments why the validity of the APP can be expected for spatially localized solutions and why it cannot be expected for non-localized solutions. Although, we restrict ourselves to a nonlinear wave equation as original system we believe that the results hold in more general situations, too.

General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations

We derive the soliton matrices corresponding to an arbitrary number of higher-order normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order solitons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann–Hilbert problem. We have applied these general results to the three-wave interaction system, and derived new classes of higher-order soliton and two-soliton solutions, in complement to those from our previous publication [Stud. Appl. Math. 110, 297 (2003)], where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically describe the simultaneous breakup of a pumping wave (u3) into the other two components (u1 and u2) and merger of u1 and u2 waves into the pumping u3 wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumping u3 wave into the u1 and u2 components, and the reverse process. In the nongeneric cases, these two-soliton solutions could describe the elastic interaction of the u1 and u2 waves, thus reproducing previous results obtained by Zakharov and Manakov [Zh. Éksp. Teor. Fiz. 69, 1654 (1975)] and Kaup [Stud. Appl. Math. 55, 9 (1976)].

Quantum Inverse Scattering Method for an Integrable Model

The exact eigenstates of the Hamiltonian for a quantum three-wave interaction system with boson-fermion operators are constructed by the quantum inverse scattering method. The Bethe-ansatz equations are obtained from the periodic boundary conditions.

Periodic and solitary wave solutions of the three-wave problem. A different approach

We consider the inverse problem for a three-wave interaction system in a manner different from Zakharovet al. and of Kaup. Our method is an adaptation of the technique due to Date to a 3 × 3 Lax pair. The analysis leads to a system of ordinary nonlinear equations for theμivariables linearizable through a suitable definition of differential on a Riemann surface. Next, in the degenerate case, when theμiare equal in pairs, we prove that such a set of equations is exactly integrable and leads to solitary solutions.

Quantum nonlinear wave interaction system

The quantum spectral transform method is applied to a Lie model (quantum three-wave interaction system). The eigenvectors of the integrals of motion are constructed using the algebraic Bethe ansatz. Different reductions and generalizations of the three wave system are presented.