Showing papers on "Cnoidal wave published in 2000"
TL;DR: In this paper, it was shown that the solutions of the water wave problem in the long-wave limit can be split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries equation.
Abstract: The Korteweg-de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two-dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long-wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.
236 citations
TL;DR: In this paper, a method to find the wave trains whose evolution leads to the freak wave formation is proposed based on the solution of the Korteweg-de-Vries equation with an initial condition corresponding to the expected freak wave.
100 citations
TL;DR: A spectral method is used to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, which there is at present no analytic proof.
Abstract: We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory.
38 citations
TL;DR: In this article, the KdV-ZK (Korteweg-de Vries-Zakharov-Kuznetsov) equations for kinetic Alfven and ion-acoustic waves in a nonthermal magnetized plasma have been derived.
Abstract: The KdV-ZK (Korteweg–de Vries–Zakharov–Kuznetsov) equations for kinetic Alfven and ion-acoustic waves in a nonthermal magnetized plasma have been derived. The coefficient of the nonlinear term of this equation for an ion-acoustic wave can vanish along a curve in a two-dimensional parameter space. In this case, two coupled equations constituting a modified KdV-ZK equation describing nonlinear behavior of an ion-acoustic wave are derived. The solitary wave solutions of all these equations are obtained and their stabilities are investigated by the Rowlands–Infeld method. It is found that the kinetic Alfven solitons are stable. The instability conditions and the maximum growth rate of the instability for ion-acoustic solitons are determined.
31 citations
TL;DR: In this article, the performance of a 1-D Boussinesq model is evaluated against laboratory data for its ability to predict surf zone velocity moments, and it is found that the model handles velocity moments better in the shorter wave tests than in the long wave cases where triad interactions are stronger.
25 citations
TL;DR: In this article, the authors used an extended BenjaminBonaMahony (eBBM) equation with higher-order nonlinear and dispersive effects, which is asymptotically equivalent to the eKdV equation.
Abstract: Solitary wave interaction is examined, for the case of surface waves on shallow water, by using an extended BenjaminBonaMahony (eBBM) equation. This equation includes higherorder nonlinear and dispersive effects, and hence is asymptotically equivalent to the extended Kortewegde Vries (eKdV) equation. However, it has certain numerical advantages as it allows the modelling of steeper waves, which, due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of collisions of varying nonlinearity are performed. The numerical results show evidence of inelastic behaviour at high order. For waves of small amplitude the evidence of inelastic behaviour is indirect; after collision a dispersive wavetrain of extremely small amplitude is found behind the smaller solitary wave. For steeper waves, however, direct evidence of inelastic behaviour is found; the larger wave is increased and the smaller wave is decreased in amplitude after the collision. Conservation laws for the ...
18 citations
TL;DR: This work considers the self-compression of the cnoidal waves of both cn- and dn-types in the materials with focusing Kerr nonlinearity and the main features of the wave propagation are analysed on the basis of finite number harmonic approximation.
18 citations
01 Jan 2000
TL;DR: The solitary wave as mentioned in this paper is a single elevation of water above the originally undisturbed level as shown in Figure 1 and it is translatory, a passing wave causing a definite net horizontal displacement of the liquid.
Abstract: The solitary wave consists of a single elevation of water above the originally undisturbed level as shown in Figure 1. It is translatory, a passing wave causing a definite net horizontal displacement of the liquid. While the characteristics of oscillatory waves depend on wave length as well as wave height and water depth, the solitary wave is apparently described completely by the wave height and water depth so long as attenuation due to friction is unimportant.
17 citations
TL;DR: In this paper, the authors analyzed the asymptotic behavior of the rescaled solution to the linear Korteweg-de Vries equation when the initial conditions are supposed to be random and weakly dependent.
Abstract: We analyze the asymptotic behavior of the rescaled solution to the linear Korteweg–de Vries equation when the initial conditions are supposed to be random and weakly dependent. By means of the method of moments we prove the Gaussianity of the limiting process and we present its correlation function. The same technique is applied to the analysis of another third-order heat-type equation.
17 citations
TL;DR: In this paper, the propagation of weakly nonlinear waves in a thin tube medium is studied through the use of the modified multiple scale expansion method, where the evolution of the lowest order (first order) term in the perturbation expansion may be described by the Korteweg-de Vries equation.
Abstract: In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, the propagation of weakly nonlinear waves, in such a medium is studied through the use of the modified multiple scale expansion method. It is shown that the evolution of the lowest order (first order) term in the perturbation expansion may be described by the Korteweg-de Vries equation. The governing equation for the second order terms and the localized travelling wave solution for these equations are also obtained. The applicability of the present model to flow problems in arteries is discussed.
13 citations
TL;DR: In this paper, a family of stationary stationary solitary wave solutions whose members are distinguished by the number of "humps" is found for a given η, corresponding to each solitary wave.
TL;DR: In this article, a new wave dispersion equation was derived, in which the soil characteristics of the porous seabed were included, and the incident wave angle of the short-crested wave and the properties of the seafloor were investigated.
TL;DR: In this paper, the accuracy of several asymptotic series expansions for wave speed and particle velocity under the crest of a solitary wave (on a fluid at rest) up to maximum height is investigated, based on a scaling of calculated properties of long periodic waves to the case of solitary waves.
TL;DR: In this paper, the authors investigated numerically the periodic and solitary wave solutions for a perturbed Kortewegde Vries equation with unstable and dissipation terms in Hilbert transform and found that the tails of these waves decay as O(1/∣x∣2) when ∣x ∣→∞ irrespective of the magnitude of η.
TL;DR: In this paper, the influence of the nonlocal component of the photorefractive crystal response on the cnoidal wave propagation dynamic was analyzed and the dependence of the self-bending parameter on the degree of the spatial localization of the wave energy was shown.
01 Jan 2000
TL;DR: In this paper, the effect of interphase heat transfer on shock wave propagation is investigated, and a multi-wave nonlinear equation which in the limiting case of the absence of heat transfer decomposes into two classic generalizations of the Boussinesq equations is derived.
Abstract: The effect of interphase heat transfer on shock wave propagation is investigated. A multiwave nonlinear equation which in the limiting case of the absence of heat transfer decomposes into two classic generalizations of the Boussinesq equations is derived. Quasi-isothermal and quasi-adiabatic propagation regimes for which the heat transfer is fairly intense are considered. For both regimes, nonlinear equations describing the wave propagation are obtained. The equation describing the first regime is investigated in detail. Exact analytic solutions of this equation are given and used to study the shock wave structures and the solitary wave behavior. Formulas for the dependence of the heat transfer rate on the equilibrium-mixture parameters are obtained.
TL;DR: In this article, instead of the strongly linear boundary-valua problem with a free boundary containing several unknown functions, we solve an ordinary quadratic-nonlinear differential-difference equation of the first order containing an unknown function.
Abstract: A solution of the problem of gravity waves on a liquid surface is sought in the form of a series whose first term corresponds to shallow-water theory. Such series have been previously studied numerically and analytically but their structure remains unclear because of the complicated initial formulation of the problem. In the present paper, instead of the strongly linear boundary-valua problem with a free boundary containing several unknown functions, we solve an ordinary quadratic-nonlinear differential-difference equation of the first order containing an unknown function.
TL;DR: In this paper, the three-dimensional power Kortewegde Vries equation (ut +u n ux +uxxx)x + uyy + uzz = 0, is considered.
Abstract: The three-dimensional power Korteweg-de Vries equation (ut +u n ux +uxxx)x + uyy + uzz = 0, is considered. Solitary wave solutions for any positive integer n and cnoidal wave solutions for n = 1 and n = 2 are obtained. The cnoidal wave solutions are shown to be represented as infinite sums of solitons by using Fourier series expansions and Poisson's summation formula.
TL;DR: In this article, the authors considered the effects of the Aharonov-Bohm flux on the soliton and showed that the width, the peak, the binding energy and the effective mass of a soliton are affected.
Abstract: Cnoidal wave solutions in the quantum lattice gas model in the presence of the Aharonov-Bohm flux in a ring are found. The energy gap, the effective mass and the binding energy of cnoidal waves are given. It is pointed out that the spin-wave-like excitation and the soliton are special cases of the cnoidal wave. Effects of the Aharonov-Bohm flux on the soliton are considered. It is shown that the width, the peak, the binding energy and the effective mass of the soliton and the energy gap are affected and a Josephson-like current is induced by the Aharonov-Bohm flux.
01 Jan 2000
TL;DR: In this paper, the propagation of internal harmonic waves in a three-layer liquid is examined in a linear approximation for arbitrary wave numbers and a traveling wave so-called dispersion equation is derived.
Abstract: The propagation o f internal harmonic waves in a three-layer liquid is examined in a linear approximation for arbitrary wave numbers. A travelling wave sohaion is obtained and a dispersion equation is derived. The roots of the dispersion equation and the dependence of the solution on the ratio of the densities of the liquids are examined in detail. The phase and group velocities are plotted as functions of wavelength and graphs of the various wave modes are presented.