Showing papers on "Cnoidal wave published in 2001"
TL;DR: In this article, a Jacobi elliptic function expansion method was proposed to construct the exact periodic solutions of nonlinear wave equations, which includes some shock wave solutions and solitary wave solutions.
1,231 citations
TL;DR: In this paper, the Jacobi elliptic functions are applied in Jacobi function expansion method to construct the exact periodic solutions of nonlinear wave equations and it is shown that more new periodic solutions can be obtained by this method and more shock wave solutions or solitary wave solution can be got at their limit condition.
509 citations
TL;DR: Based upon the Riccati equation, a new generalized transformation is presented and applied to solve Whitham-Broer-Kaup (WBK) equation in shallow water as mentioned in this paper.
335 citations
TL;DR: In this article, a Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation is derived for a plasma comprised of cool and hot electrons and a species of fluid ions.
Abstract: Motivated by a recent paper [Phys. Plasmas 7, 2987 (2000)] highlighting the potential importance of the electron-acoustic wave in interpreting the solitary waves observed by high time resolution measurements of the electric field in the auroral region, the effect of a magnetic field on weakly nonlinear electron-acoustic waves is investigated. A Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation is derived for a plasma comprised of cool and hot electrons and a species of fluid ions. Two models are employed for the ions: magnetized and unmagnetized. When the ions are magnetized the frequency constraints imposed upon the electron-acoustic wave packet prove to be too limiting to be of general use. The second model, which treats the ions as a stationary neutralizing background, overcomes the restrictions imposed by the former and is more fitting for the frequency domain of the electron-acoustic wave. Plane and ellipsoidal soliton solutions are admitted by the KdV-ZK equation, the latter perhaps able to explain some of the two dimensional features of the solitary waves observed in the Earth’s high altitude auroral region. Both models for the ions predict only negative potential solitons. It is discussed how the plasma model might be adapted to produce positive potential solitons.
180 citations
TL;DR: In this paper, the dynamics of wave groups are studied for long waves, using the framework of the extended Korteweg-de Vries equation, and it is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term, and for a high carrier frequency.
102 citations
TL;DR: In this article, a Jacobi elliptic function expansion method was proposed to construct the exact periodic solutions of nonlinear wave equations, including the shock wave solutions and the solitary wave solutions.
Abstract: A Jacobi elliptic function expansion method is proposed to construct the exact periodic solutions of nonlinear wave equations. This new method contains the hyperbolic tangent expansion method, and the periodic solutions obtained by this method, include the shock wave solutions and the solitary wave solutions.
76 citations
TL;DR: The noninertia wave is one of the special cases of the diffusion wave as mentioned in this paper, which is the wave whose induced disturbance in flow is analogous to the diffusion of particles or heat.
61 citations
TL;DR: In this paper, a nonlinear wave equation derived from a simplified liquid crystal model is studied, in which the wave speed is a given function of the wave amplitude, and a viscous approximation of the equilibria is derived.
Abstract: We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equ...
59 citations
56 citations
TL;DR: In this article, the problem of tsunami wave generation by variable meteo-conditions is discussed, and the simplified linear and nonlinear shallow water models are derived, and their analytical solutions for a basin of constant depth are discussed.
Abstract: . The problem of tsunami wave generation by variable meteo-conditions is discussed. The simplified linear and nonlinear shallow water models are derived, and their analytical solutions for a basin of constant depth are discussed. The shallow-water model describes well the properties of the generated tsunami waves for all regimes, except the resonance case. The nonlinear-dispersive model based on the forced Korteweg-de Vries equation is developed to describe the resonant mechanism of the tsunami wave generation by the atmospheric disturbances moving with near-critical speed (long wave speed). Some analytical solutions of the nonlinear dispersive model are obtained. They illustrate the different regimes of soliton generation and the focusing of frequency modulated wave packets.
35 citations
TL;DR: The hyperbolic function method for nonlinear wave equations is presented in this paper, which is based on the fact that the solitary wave solutions are essentially of a localized nature and can be written as polynomials of hyperbola functions.
Abstract: The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grobner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.
TL;DR: In this article, the authors present an algorithm for solving the initial value problem on the infinite line for a wave equation arising in the study of long waves at the free surface of water, exploiting the link between the hierarchy of the shallow water equation and that of the modified Korteweg-de Vries equation.
TL;DR: In this paper, experimental evidence of the cnoidal waves self-compression in photorefractive BTO crystal with drift nonlinearity is reported. But the selfcompression processes for a high voltages are limited by development of dynamic instability.
TL;DR: In this article, the scattering of a soliton from a cnoidal wave train in a fiber theoretically as well as numerically is analyzed, and the effect of collisions is described by the change of velocities of solitons, and theoretical predictions are in good agreement with numerical results.
Abstract: We analyze the scattering of a soliton from a cnoidal wave train in a fiber theoretically as well as numerically. Solitons recover their original shapes and velocities after collisions, while shapes of cnoidal waves are nearly preserved during collisions. The effect of collisions is described by the change of velocities of solitons, and the theoretical predictions are in good agreement with numerical results.
TL;DR: In this article, the propagation of weakly nonlinear waves in a thin tube medium is studied through the use of the modified multiple expansion method, where the evolution of the lowest order (first-order) term in the perturbation expansion may be described by the Korteweg-de Vries (KdV) equation.
TL;DR: In this article, 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 Alfven wave in one Cartesian space dimension.
Abstract: 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 Alfven wave in one Cartesian space dimension. The large-amplitude Alfven 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 Alfven 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, w) = 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 w = w(k), where k is real, and in the form k = k(w), where w 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 Schrodinger 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.
TL;DR: In this article, the authors compare experiments on short gravity wave phase shifting by surface solitary waves to a Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) refraction theory.
Abstract: In this paper, we compare experiments on short gravity wave phase shifting by surface solitary waves to a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) refraction theory. Both weak interactions (head-on interaction) and strong interactions (overtaking interaction) are examined. We derive a dispersion relation and wave action conservation relation which are similar to the ones obtained for short waves refraction on slowly varying media. The model requires an exact solitary wave solution. To this end, a steady wave solution is numerically computed using the algorithm devised by Byatt-Smith [Proc. R. Soc. London, Ser. A 315, 405 (1970)]. However, two other solitary wave solutions are incorporated in the model, namely the classical Korteweg and De Vries (KdV) [Phil. Mag. 39, 422 (1895)] solution (weakly nonlinear/small amplitude solitary wave) and the Rayleigh [Phil. Mag. 1, 257 (1876)] solution (strongly nonlinear/large amplitude solitary wave). Measurements of the short wave phase shift show better agreement with the theoretical predictions based on the Byatt-Smith numerical solution and the Rayleigh solution rather than the Korteweg and De Vries one for large amplitude solitary waves. Theoretical phase shifts predictions based on Rayleigh and Byatt-Smith numerical solutions agree with the experiments for A/h0⩽0.5. A new heuristic formula for the phase shift allowing for large amplitude solitary waves is proposed as a limiting case when the short wave wave number increases.
TL;DR: In this paper, a numerical method is described that may be used to determine the propagation characteristics of weakly non-hydrostatic non-linear free surface waves over a general, bottom topography.
Abstract: A numerical method is described that may be used to determine the propagation characteristics of weakly non-hydrostatic non-linear free surface waves over a general, bottom topography. In shallow water of constant undisturbed depth, such waves are equivalent to the familiar cnoidal waves characterized by sharp crests and relatively flat troughs. For a certain range of parameters, these propagate without change of form by virtue of the weakly non-hydrostatic balance in the vertical momentum equation. Effectively, this counters the tendency for the non-linearity in a purely hydrostatic theory to lead to a continuously deforming surface wave profile. The realistic representation furnished by cnoidal wave theory of free surface waves in the shallow near-shore zone has led to its utilization in evaluating their propagation characteristics. Nonetheless, the classic analytical theory is inapplicable to the case of wave propagation over a sloping beach or off-shore sand bar topography. Under these conditions, a local change in form of the surface wave profile is anticipated before the waves break and knowing this is required in order to evaluate fully the propagation process
TL;DR: In this paper, the influence of a sufficiently nonstationary Raman contribution to nonlinear susceptibility on the dynamics of cnoidal wave propagation in optical fibers is theoretically analyzed and the dependence of the parameter that describes the curvature of the propagation trajectory on the degree of localization of the wave energy is presented.
Abstract: The influence of a sufficiently nonstationary Raman contribution to nonlinear susceptibility on the dynamics of cnoidal wave propagation in optical fibers is theoretically analyzed. The dependence of the parameter that describes the curvature of the propagation trajectory on the degree of localization of the wave energy is presented.
TL;DR: Two kinds of analytic singular solutions of two classical wave equations of the Boussinesq equation and a generalized Korteweg-de Vries equation are obtained by means of the improved homogeneous balance method and a nonlinear transformation, showing that special singular wave patterns exist in the classical models of shallow water wave problem.
Abstract: In this paper, two kinds of analytic singular solutions (finite-time and infinite-time singular solutions) of two classical wave equations (the Boussinesq equation and a generalized Korteweg-de Vries equation) are obtained by means of the improved homogeneous balance method and a nonlinear transformation. The solutions show that special singular wave patterns exist in the classical models of shallow water wave problem.
TL;DR: In this article, the propagation of periodic cnoidal waves in a medium with a saturable nonlinear response is investigated, and the ranges of existence of soliton solutions are determined.
Abstract: Propagation of periodic cnoidal waves in a medium with a saturable nonlinear response is investigated. The ranges of existence of soliton solutions are determined, and the specific features of such solutions are analysed within the limiting cases of strong and weak energy localisation in cnoidal waves. Optical multistability is discussed in the context of the existence of periodic solutions. The stability of cnoidal waves is studied numerically and analytically.
TL;DR: In this article, the propagation of weakly non-linear waves in a prestressed thin elastic tube filled with an incompressible layered fluid, where the outer layer is assumed to be inviscid whereas the cylindrical core is considered to be viscous, was studied using reductive perturbation technique.
01 Jan 2001
TL;DR: In this article, a modified Smagorinsky subgrid-scale model is used to describe the turbulence effect in a wave propagation or shoaling and breaking process, and the model results are in good agreement with analytical solution and experimental data.
Abstract: In this paper, the large eddy simulation method is used combined with the marker and cell method to study the wave propagation or shoaling and breaking process. As wave propagates into shallow water, the shoaling leads to the increase of wave height, and then at a certain position, the wave will be breaking. The breaking wave is a powerful agent for generating turbulence, which plays an important role in most of the fluid dynamic processes throughout the sarf zone, such as transformation of wave energy, generation of near-shore current and diffusion of materials. So a proper numerical model for describing the turbulence effect is needed. In this paper, a revised Smagorinsky subgrid-scale model is used to describe the turbulence effect. The present study reveals that the coefficient of the Smagorinsky model for wave propagation or breaking simulation may be taken as a varying function of the water depth and distance away from the wave breaking point. The large eddy simulation model presented in this paper has been used to study the propagation of the solitary wave in constant water depth and the shoaling of the non-breaking solitary wave on a beach. The model is based on large eddy simulation, and to track free-surface movements, the Tokyo University Modified Marker and Cell (TUMMAC) method is employed. In order to ensure the accuracy of each component of this wave mathematical model,several steps have been taken to verify calculated solutions with either analytical solutions or experimental data. For non-breaking waves, very accurate results are obtained for a solitary wave propagating over a constant depth and on a beach. Application of the model to cnoidal wave breaking in the surf zone shows that the model results are in good agreement with analytical solution and experimental data. From the present model results, it can be seen that the turbulent eddy viscosity increases from the bottom to the water surface in surf zone. In the eddy viscosity curve, there is a turn-point obviously, dividing water depth into t
TL;DR: In this paper, an action balance equation model is proposed to calculate the wave height and wave period field in the Haian Bay area and to simulate the influences of the unsteady current and water level variation on the wave field.
Abstract: Several current used wave numerical models are briefly described, the computing techniques of the source terms, numerical wave generation and boundary conditions in the action balance equation model are discussed. Not only the quadruplet wave-wave interactions, but also the triad wave-wave interactions are included in the model, so that nearshore waves could be simulated reasonably. The model is compared with the Boussinesq equation and the mild slope equation. The model is applied to calculating the distributions of wave height and wave period field in the Haian Bay area and to simulating the influences of the unsteady current and water level variation on the wave field. Finally, the developing tendency of the model is discussed.
01 Jan 2001
TL;DR: In this paper, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation, and then a practical method for the simulation of wave height and wave set-up in nearshore regions is presented.
Abstract: Based on the time dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation, and then a practical method for the simulation of wave height and wave set- up in nearshore regions is presented. The variation of the complex wave amplitude is numerically simulated by use of the parabolic mild slope equation including the effect of wave energy dissipation due to wave breaking. The components of wave radiation stress are calculated subsequently by new expressions for them according to the obtained complex wave amplitude, and then the depth-averaged equation is applied to the calculation of wave set-up due to wave breaking. Numerical results are in good agreement with experimental data, showing that the expression for the energy dissipation factor is reasonable and that the new method is effective for the simulation of wave set-up due to wave breaking in nearshore regions.
01 Jan 2001
TL;DR: In this paper, the nonexistence of global solutions to wave equations of the type utt − ∆u ± ut = λu + |u| 1+q is considered.
Abstract: In this paper the nonexistence of global solutions to wave equations of the type utt − ∆u ± ut = λu + |u| 1+q is considered. We derive, for an averaging of solutions, a nonlinear second differential inequality of the type w ± w ≥ bw + |w|, and we prove a blowing up phenomenon under some restriction on u(x, 0) and ut(x, 0). Similar results are given for other equations. 1 – Introduction In [2] Glassey proved the non global existence of classical solutions to
TL;DR: In this paper, conditions of the stable balance between competing processes of the Raman self-frequency shift and bandwidth-limited amplification of the periodical cnoidal waves were investigated.
01 Jan 2001
TL;DR: In this article, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation, and then a practical method for the simulation of wave height and wave set-up in nearshore regions is presented.
Abstract: Based on the time dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation, and then a practical method for the simulation of wave height and wave set-up in nearshore regions is presented. The variation of the complex wave amplitude is numerically simulated by use of the parabolic mild slope equation including the effect of wave energy dissipation due to wave breaking. The components of wave radiation stress are calculated subsequently by new expressions for them according to the obtained complex wave amplitude, and then the depth-averaged equation is applied to the calculation of wave set-up due to wave breaking. Numerical results are in good agreement with experimental data,showing that the expression for the energy dissipation factor is reasonable and that the new method is effective for the simulation of wave set-up due to wave breaking in nearshore regions.
TL;DR: In this paper, specific features of SRS conversion of multisoliton periodic cnoidal waves into high-energy Cnoidal pump wave at the Stokes frequency falling within the range of anomalous group-velocity dispersion are investigated.
Abstract: Specific features of SRS conversion of multisoliton periodic cnoidal waves into high-energy cnoidal waves at the Stokes frequency falling within the range of anomalous group-velocity dispersion are investigated Dependences of the maximum compression degree of the pump pulse and the characteristic distance of Stokes-wave excitation on the energy localisation parameter of the cnoidal pump wave are presented
TL;DR: In this paper, the generalized KdV equation in (2+1)-dimensional space arising from the multidimensional isospectral flows associated with the second-order scalar operators was obtained by using the direct method.
Abstract: In this paper,eight types of (1+1)-dimensional similarity reductions which contain variable coefficient equation,are obtained for the generalized KdV equation in (2+1)-dimensional space arising from the multidimensional isospectral flows associated with the second-order scalar operators by using the direct method.In addition,the cnoidal wave solution and dromion-like solution are also derived by using the reduced nonlinear ordinary differential equations.The (1+1) dromion obtained by Lou [J.Phys.A28 (1995) 7227] and Zhang [Chin.Phys.9 (2000) 1] is only a special case of our results.Moreover,some properties of the dromion-like solutions are analyzed.