Wave Interaction Equations Research Articles

A numerical study of the long wave–short wave interaction equations

Two numerical methods are presented for the periodic initial-value problem of the long wave–short wave interaction equations describing the interaction between one long longitudinal wave and two short transverse waves propagating in a generalized elastic medium. The first one is the relaxation method, which is implicit with second-order accuracy in both space and time. The second one is the split-step Fourier method, which is of spectral-order accuracy in space. We consider the first-, second- and fourth-order versions of the split-step method, which are first-, second- and fourth-order accurate in time, respectively. The present split-step method profits from the existence of a simple analytical solution for the nonlinear subproblem. We numerically test both the relaxation method and the split-step schemes for a problem concerning the motion of a single solitary wave. We compare the accuracies of the split-step schemes with that of the relaxation method. Assessments of the efficiency of the schemes show that the fourth-order split-step Fourier scheme is the most efficient among the numerical schemes considered.

Propagation of Alfvén waves in shear flows: Nature of driven longitudinal velocity and density fluctuations

The nature of driven longitudinal velocity fluctuations and density fluctuations is studied when there is an Alfvén wave propagating in the medium characterized by an inhomogeneous velocity field. A system of governing equations that describes driven fluctuations is obtained and the properties of this system are analyzed. For some circumstances considering general properties of the Alfvén wave, the system allows us to obtain analytical solutions for driven waves that can be compared with ones obtained by numerical integration of the wave interaction equations. It is shown that for the low β plasma, the analytical solutions are in a good agreement with the numerical ones. The natural frequencies of individual wave components (“uncoupled natural frequencies”) are analyzed and how these frequencies are related to the natural frequencies of the coupled system are shown. In the summary, it is discussed where this process can be important. The possible observations in different areas where its signature can be found are outlined.

New Exact Solutions to Long-Short Wave Interaction Equations

New exact solutions expressed by the Jacobi elliptic functions areobtained to the long-short wave interaction equations by using the modifiedF-expansion method. In the limit case, solitary wave solutions andtriangular periodic wave solutions are obtained as well.

Zero-dispersion limit of the short-wave–long-wave interaction equations

The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient λ is small order of ε, λ = o ( ε ) , the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation.

Magnetohydrodynamic waves in non-uniform flows I: a variational approach

The interaction of magnetohydrodynamic (MHD) waves in a non- uniform, time-dependent background plasma flow is investigated using Lagrangian field theory methods. The analysis uses Lagrangian maps, in which the position of the fluid element x ∗ is expressed as a vector sum of the position vector x of the background plasma fluid element plus a Lagrangian displacement ξ(x ,t ) due to the waves. Linear, non-Wentzel-Kramer-Brillouin (WKB) wave interaction equations are obtained by expansion of the Lagrangian out to second order in ξ and ∆S, where ∆S is the Lagrangian entropy perturbation. The characteristic manifolds of the waves are determined by consideration of the Cauchy problem for the wave interaction equations. The manifolds correspond to the usual MHD waves modes, namely the Alfvwaves, the fast and slow magnetoacoustic waves and the entropy wave. The relationships between the characteristic manifolds, and the ray equations of geometrical MHD optics are developed using the theory of Cauchy characteristics for first-order partial differential equations. The first-order differential equations describing the singular manifolds are the dispersion equations for the MHD ei- genmodes, where the wave vector k = ∇φ and frequency ω = −φt correspond to the characteristic manifolds φ(x ,t )= constant. The form of the characteristic manifolds for both time-dependent and steady MHD flows are developed. The bi- characteristics for steady MHD waves in a steady background flow are related to the group velocity surface and Mach cone for the waves, and determine when the flow is elliptic, hyperbolic, or of mixed hyperbolic-elliptic type. The wave interaction equations are decomposed into coupled equations for the compressible and incompressible perturbations.

Pfaffianization of the discrete three-dimensional three wave interaction equation

We first present the discrete Gram-type determinant solution to the discrete three-dimensional three wave interaction equation. Then a pfaffianized version of the discrete three-dimensional three wave interaction equation is produced by using Hirota and Ohta’s pfaffianization procedure. The Gram-type pfaffian solution to the resulting system is also given. In addition, we obtain the soliton solutions of the resulting system as an example of the Gram-type pfaffian solution.

Resonant interaction between gravitational waves, electromagnetic waves, and plasma flows

In magnetized plasmas gravitational and electromagnetic waves may interact coherently and exchange energy between themselves and with plasma flows. We derive the wave interaction equations for these processes in the case of waves propagating perpendicular or parallel to the plasma background magnetic field. In the latter case, the electromagnetic waves are taken to be circularly polarized waves of arbitrary amplitude. We allow for a background drift flow of the plasma components which increases the number of possible evolution scenarios. The interaction equations are solved analytically and the characteristic time scales for conversion between gravitational and electromagnetic waves are found. In particular, it is shown that in the presence of a drift flow there are explosive instabilities resulting in the generation of gravitational and electromagnetic waves. Conversely, we show that energetic waves can interact to accelerate particles and thereby \emph{produce} a drift flow. The relevance of these results for astrophysical and cosmological plasmas is discussed.

Jacobi elliptic function expansion method applied to long-short wave interaction equations

In this paper,the Jacobi elliptic function method is generalized to study nonlinear evolution system. The envelope periodic solutions of long-short wave interaction equation are obtained. The solitary wave solutions for this model are also given under some conditions.

Parametric instabilities and wave coupling in Alfvén simple waves

A wave coupling formalism for magnetohydrodynamic (MHD) waves in a non-uniform background flow is used to study parametric instabilities of the large-amplitude, circularly polarized, simple plane Alfvén wave in one Cartesian space dimension. The large-amplitude Alfvén wave (the pump wave) is regarded as the background flow, and the seven small-amplitude MHD waves (the backward and forward fast and slow magnetoacoustic waves, the backward and forward Alfvén waves, and the entropy wave) interact with the pump wave via wave coupling coefficients that depend on the gradients and time dependence of the background flow. The dispersion equation for the waves D(k,ω) = 0 obtained from the wave coupling equations reduces to that obtained by previous authors. The general solution of the initial value problem for the waves is obtained by Fourier and Laplace transforms. The dispersion function D(k,ω) is a fifth-order polynomial in both the wavenumber k and the frequency ω. The regions of instability and the neutral stability curves are determined. Instabilities that arise from solving the dispersion equation D(k,ω) = 0, both in the form ω = ω(k), where k is real, and in the form k = k(ω), where ω is real, are investigated. The instabilities depend parametrically on the pump wave amplitude and on the plasma beta. The wave interaction equations are also studied from the perspective of a single master wave equation, with multiple wave modes, and with a source term due to the entropy wave. The wave hierarchies for short- and long-wavelength waves of the master wave equation are used to discuss wave stability. Expanding the dispersion equation for the different long-wavelength eigenmodes about k = 0 yields either the linearized Korteweg–deVries equation or the Schrödinger equation as the generic wave equation at long-wavelengths. The corresponding short-wavelength wave equations are also obtained. Initial value problems for the wave interaction equations are investigated. An inspection of the double-root solutions of the dispersion equation for k, satisfying the equations D(k,ω) = 0 and ∂D(k,ω) = ∂k = 0 and pinch point analysis shows that the solutions of the wave interaction equations for hump or pulse-like initial data develop an absolute instability. Fourier solutions and asymptotic analysis are used to study the absolute instability.

Finite-dimensional integrable systems related to the n-wave interaction equations

Under a constraint between the potentials and the eigenfunctions, Lax pairs and adjoint Lax pairs of a soliton hierarchy associated with the n×n generalized Zakharov–Shabat eigenvalue problem are transformed into a spatial finite-dimensional Hamiltonian system and a hierarchy of temporal finite-dimensional Hamiltonian systems. The Lax representations, r-matrix structure and integrals of motion are explicitly presented. These integrals of motion are functionally independent and in involution in pairs, which shows that these systems, especially the whole hierarchy of temporal finite-dimensional Hamiltonian systems, are Liouville integrable.

Nonlinear interaction between long and short waves in a generalized elastic solid

In the present study, nonlinear interaction between long and short waves propagating in a generalized elastic medium is examined. In particular, the case where the phase velocity of the long longitudinal wave is equal to the group velocity of the short transverse waves is studied. By using an asymptotic expansion method, three coupled nonlinear evolution equations are derived for the description of the interaction. In the absence of one of the transverse waves, these equations reduce to the so-called long wave–short wave interaction equations which are also known as Zakharov–Benney (ZB) equations. Furthermore, the nonlinear interaction between a long longitudinal wave and a short longitudinal wave is considered and ZB equations are derived for the description of interaction. Some special solutions to the interaction equations are also presented.

Bilinear identities and discrete equations

Discrete counterparts to integrable systems such as the KP equation and the three-dimensional three wave interaction equations are obtained from bilinear identities which result from expanding typical solutions which appear in the form of Casorati and multi-Casorati determinants.

`Positon' and `Dromion' Solutions of the (2+1) Dimensional Long Wave-Short Wave Resonance Interaction Equations

`Positon' and `dromion' solutions are derived for the long wave-short wave interaction equations in a two-layer fluid. Positons are new, exact solutions of nonlinear evolution equations that exhibit algebraic decay in the far field. A positon solution can be generated by taking a special limit of multi-soliton expansion. Variation of the limiting process yields different solutions. Dromions are exact, localized solutions of (2+1) dimensional (2 spatial, 1 temporal) nonlinear evolution equations that decay exponentially in all directions. One and higher dromion solutions are investigated, and a particular case of higher dromion solutions is considered in details. By applying another limiting procedure a new solution is generated. This method of `coalescence of eigenvalues' or `wavenumbers' is thus quite universal, and can be applied to a wide variety of expansions, not just the multi-soliton type. Finally, the case of propagation solutions on a continuous wave background is also studied.

Wave interactions in magnetohydrodynamics, and cosmic-ray-modified shocks

Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfvén waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmic-ray-modified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfvén waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with non-zero field aligned electric current and/or vorticity (i.e. if B·∇×B≠0 and/or B·∇×u≠0, where B and u are the magnetic field induction and fluid velocity respectively).

Global well-posedness and conservation laws for the water wave interaction equation

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem foris locally well posed in the largest space where the three conservationscan be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

Painlevé Analysis of the Long and Short Wave Interaction Equation

The integrability of the long and short wave interaction equation in a coupled form of the Nonlinear Schrodinger equation and the Korteweg-de Vries equation is examined by means of the Painleve test. In particular, the cases when the reduced ordinary differential equation obtained through the traveling wave transformation has the Painleve property are examined. As a result of this, except for the special cases, it is found that this equation does not pass the test without the imposition of restrictions. However, this test is not successful in the nearly integrable case, since the singular manifold expansion becomes non-uniformly valid. Furthermore, for the particular case, the condition that permits the Painleve property is shown to be relaxed considerably for a finite time.

Local nonintegrability of long-short wave interaction equations

The algebra of higher symmetries and the space of conservation laws for Zakharov’s nonlinear equations of the interaction between long and short waves are completely described. The scheme of computations due to Vinogradov is used. As a result, the local nonintegrability of these equations is proved.

Bethe States for the Quantum Three Wave Interaction Equation

The quantum three wave interaction equation is introduced. The exact eigen-states of the Hamiltonian are constructed. Furthermore, the existence of the bound states is shown. It is concluded that the quantum three wave interaction equation is completely integrable.

An extended Konno-Wadati Formalism and Some New Backlund Transformations

We here present a simple formalism for the computation of the Backlund transformation. It is shown that the method works well with a nonlinear realization of the Lie algebra underlying the linear equation in prolongation space. The method is a simple extension of an elegant analysis of Konno and Wadati. We have computed the BT's in the cases of SIT, long-short wave interaction, Langmuir solitons and KdV equation. We have presented our formalism in the language of differential forms but the stress is on the actual computation than on the formalism.

Oceanic Internal Waves Are Not Weak Waves

Abstract It is shown that the oceanic internal wave field is too energetic by roughly two orders of magnitude to be treated theoretically as an assemblage of weakly interacting waves. This may be seen both from recent weak wave theoretical calculations which contradict their premises and also from inspection of magnitudes of advection and wave propagation terms. Thus, much recent discussion of results of implications of weak wave theory should be questioned critically. Scaling arguments based on buoyant turbulence are reviewed briefly. The role of vertical mass flux as distinguishing weak wave interactions from stronger turbulence is discussed. Possible progress by renormalization of wave interaction equations is considered.