Showing papers on "Cnoidal wave published in 2002"
TL;DR: In this article, the Jacobi elliptic function method with symbolic computation is extended to special-type nonlinear equations for constructing their doubly periodic wave solutions, such as the coupled Schrodinger-KdV equation.
308 citations
TL;DR: In this paper, the sn-and cn-function methods for finding nonsingular periodic-wave solutions to nonlinear evolution equations are described in a form suitable for automation, where sn and cn are the elliptic Jacobi snoidal andcnoidal functions, respectively.
256 citations
TL;DR: In this paper, the amplitude of the largest non-breaking wave in a shallow, stratified ocean has been investigated and it was shown that the maximum wave amplitude is given by one of three possibilities: the onset of wave breaking, the conjugate flow amplitude or a failure of the wave calculating algorithm to converge.
Abstract: In this paper we consider what effect the presence of a nonconstant background current has on the properties of large, fully nonlinear solitary internal waves in a shallow, stratified ocean. In particular, we discuss how the amplitude of the largest nonbreaking wave that it is possible to calculate depends on the background current as well as the nature of the upper bound. We find that the maximum wave amplitude is given by one of three possibilities: The onset of wave breaking, the conjugate flow amplitude or a failure of the wave calculating algorithm to converge (associated with shear instability). We also discuss how wave properties such as propagation speed, half-width, etc. vary with background current amplitude.
144 citations
TL;DR: In this paper, a higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density-and current-stratified shear flow with a free surface.
Abstract: . A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.
128 citations
TL;DR: In this paper, the authors used the phase plane to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations and obtained all possible phase portraits in the parametric space for the traveling wave systems.
Abstract: The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations. By using the bifurcation theory of dynamical systems to do qualitative analysis, all possible phase portraits in the parametric space for the traveling wave systems are obtained. It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied. The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.
102 citations
TL;DR: It is shown that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a "sech"-profile.
Abstract: We study the extended Korteweg–de Vries equation, that is, the usual Korteweg–de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg–de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude “table-top” solitons as well as small-amplitude solitons similar to the solitons of the Korteweg–de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a “sech”-profile.
92 citations
TL;DR: In this article, the authors decompose the solution of a coupled system into solving a set of algebraic equations and a first order ordinary differential equation of a certain unknown function, which gives us a clear relation between unknown functions for a nonlinear coupled system, and easily provides us with a series of new and more general travelling wave solutions in terms of special functions such as hyperbolic, rational, triangular, Weierstrass and Jacobi double periodic functions.
84 citations
TL;DR: In this paper, a new approach is proposed to construct exact periodic solutions to nonlinear sine-Gordon equations based on new transformations from the sine Gordon equation, based on these transformations, the authors show that more new periodic solutions can be obtained by this new approach and more shock wave solutions or solitary wave solutions under their limit condition.
79 citations
TL;DR: In this paper, a weakly nonlinear analysis of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, is presented.
Abstract: Nonlinear waves in a forced channel flow are considered The forcing is due to a bottom obstruction The study is restricted to steady flows A weakly nonlinear analysis shows that for a given obstruction, there are two important values of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, FC>1 and FL FC, there are two symmetric solitary waves sustained over the site of forcing, and at F=FC the two solitary waves merge into one; (iv) when F>FC, there is also a one-parameter family of solutions matching the upstream (supercritical) uniform flow with a cnoidal wave downstream; (v) for a particular value of F>FC, the downstream wave can be eliminated and the solution becomes a reversed hydraulic fall (it is the same as solution (ii), except that the flow is reversed!) Flows of type (iv), including the hydraulic fall (v) as a special case, are computed here using the full Euler equations The problem is solved numerically by a boundary-integral-equation method due to Forbes and Schwartz It is confirmed that there is a three-parameter family of solutions with a train of waves downstream The three parameters can be chosen as the Froude number, the obstruction size and the wavelength of the downstream waves This three-parameter family differs from the classical two-parameter family of subcritical flows (i) but includes as a particular case the hydraulic falls (ii) or equivalently (v) computed by Forbes
70 citations
TL;DR: In this paper, a weakly nonlinear analysis of the interaction between a water wave and a floating ice cover in river channels is presented, where the ice cover is assumed to be a thin uniform elastic plate.
Abstract: A nonlinear analysis of the interaction between a water wave and a floating ice cover in river channels is presented. The one-dimensional weakly nonlinear equation for shallow water wave propagation in a uniform channel with a floating ice cover is derived. The ice cover is assumed to be a thin uniform elastic plate. The weakly nonlinear equation is a fifth-order KdV equation. Analytical solutions of the nonlinear periodic wave equation are obtained. These solutions show that the shape, wavelength and celerity of the nonlinear waves depend on the wave amplitude. The wave celerity is slightly smaller than the open water wave celerity. The wavelength decreases as the wave amplitude increases. Based on these solutions the fracture of the ice cover is analysed. The spacing between transverse cracks varies form 50 m to a few hundred metres with the corresponding wave amplitude varying from 0.2 to 0.8 m, depending on the thickness and strength of the cover. These results agree well with limited field observations.
53 citations
TL;DR: This paper shows that suitable linear combinations of known periodic traveling wave solutions involving Jacobi elliptic functions yield many additional solutions with different periods and velocities.
Abstract: Even though the KdV and modified KdV equations are nonlinear, we show that suitable linear combinations of known periodic solutions involving Jacobi elliptic functions yield a large class of additional solutions. This procedure works by virtue of some remarkable new identities satisfied by the elliptic functions.
TL;DR: In this article, the initial boundary value problem for the Korteweg-de Vries (KdV) equation on the negative quarter-plane, x 0, is considered.
Abstract: The initial boundary–value problem for the Korteweg–de Vries (KdV) equation on the negative quarter–plane, x 0, is considered. The formulation of this problem is different to the usual initial boundary–value problem on the positive quarter–plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter–plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter–plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to the negative quarter–plane problem. Five qualitatively different types of solution are found, depending on the relation between the initial and boundary values. Excellent comparisons are obtained between these solutions and full numerical solutions of the KdV equation.
TL;DR: In this paper, second and third order approximations of water wave equations of the Korteweg-de Vries (KdV) type are studied and the form of the solitary wave and its amplitude-velocity dependence are identical to the sech2 formula of the one-soliton solution of the kdV.
Abstract: In this work we study second and third order approximations of water wave equations of the Korteweg–de Vries (KdV) type. First we derive analytical expressions for solitary wave solutions for some special sets of parameters of the equations. Remarkably enough, in all these approximations, the form of the solitary wave and its amplitude-velocity dependence are identical to the sech2 formula of the one-soliton solution of the KdV. Next we carry out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finite-difference scheme, in parameter regions where soliton-like behavior is observed. In these regions, we find solitary waves which are stable and behave like solitons in the sense that they remain virtually unchanged under time evolution and mutual interaction. In general, these solutions sustain small oscillations in the form of radiation waves (trailing the solitary wave) and may still be regarded as stable, provided these radiation waves do not exceed a nume...
TL;DR: In this article, a forced extended Korteweg-de Vries model is used to study the turbulent flow of a two-layer fluid over an obstacle. But the model is not suitable for the case when the layer depths are near critical.
Abstract: Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg–de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg–de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion ter...
TL;DR: In this article, the Tanh method is proposed to find travelling wave solutions in (1+1 and (2+1) dimensional wave equations, and it can be extended to solve a whole family of modified Korteweg-de Vries type of equations, higher dimensions wave equations and nonlinear evolution equations.
TL;DR: The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg as mentioned in this paper.
Abstract: The existence of a line solitary-wave solution to the water-wave problem with strong surface-tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchgassner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Korteweg-de Vries line solitary wave belongs to a family of periodically modulated solitary waves which have a solitary-wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. It is shown that the Korteweg-de Vries solitary wave undergoes a dimension-breaking bifurcation that generates a family of periodically modulated solitary waves. The term dimension-breaking phenomenon describes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.
TL;DR: In this paper, a generalized conditional symmetry approach for the functional separation of variables in a non-linear wave equation with a nonlinear wave speed was developed, which was used to obtain a number of new (1+1)-dimensional nonlinear Wave equations with variable wave speeds admitting a functionally separable solution.
Abstract: We develop a generalized conditional symmetry approach for the functional separation of variables in a nonlinear wave equation with a nonlinear wave speed. We use it to obtain a number of new (1+1)-dimensional nonlinear wave equations with variable wave speeds admitting a functionally separable solution. As a consequence, we obtain exact solutions of the resulting equations.
TL;DR: In this paper, simple explicit solutions of the form (1) of the wave equation with three space variables x1, x2, and x3 were given. But they were not given for the case where the phase θ and the distortion factor g are given functions of the space variables and time and the function f(θ) describing the wave shape.
Abstract: of hyperbolic equations, where the phase θ and the distortion factor g are given functions of the space variables and time and the function f(θ) describing the wave shape is arbitrary. The examples of the plane wave with θ = x1−ct and g = 1 and the spherical wave with θ = |x|−ct and g = |x|−1, where |x| = (x1 + x2 + x3) , were given in [1] for the wave equation with three space variables x1, x2, and x3. We are interested in simple explicit solutions of the form (1) of the wave equation
TL;DR: In this paper, several finite-volume schemes are developed and applied to simulate nonlinear, dispersive, unidirectional waves propagating over a flat bed, and three methods are used to discretize the advection portion of the governing equation.
Abstract: Several finite-volume schemes are developed and applied to simulate nonlinear, dispersive, unidirectional waves propagating over a flat bed. These schemes differ mainly in the treatment of advection, while dispersion is treated the same among the different models. Three methods—linear, total variation diminishing, and essentially nonoscillatory—are used to discretize the advection portion of the governing equation. The linear schemes are analyzed with Von Neumann's method to discern stability limits as well as their damping and dispersion characteristics. In addition, predictions from all of the models are compared with analytical solutions for solitary and cnoidal waves as well as experimental data for undular bores. The finite-volume methods are also compared with a second-order accurate finite-difference scheme. The results indicate that the finite-volume schemes yield more accurate solutions than the finite-difference scheme. In addition, the Warming-Beam and Fromm linear finite-volume schemes yielded the most accurate solutions and were among the most computationally efficient schemes tested.
TL;DR: In this paper, it was shown that the nonlinear dust acoustic wave can be described by the modified Korteweg-de Vries equation under high-order transverse perturbations.
Abstract: By considering the vortex-like ion distribution, we know that the nonlinear dust acoustic wave can be described by the modified Korteweg-de Vries equation. If there are high-order transverse perturbations in this system, the governing equation for this system is the modified Kadomtsev-Petviashvili equation. We also find that for this system under the transverse perturbation the one-dimensional solitary wave solution is stable.
TL;DR: In this paper, five methods for coping with singular systems are described: (i) reformulation of the problem, (ii) deleting residuals, (iii) Keller's bordered matrix scheme, (iv) QR-factored, overdetermined Newton iteration, and (v) pseudoinverse Newton iteration.
TL;DR: In this article, a cnoidal wave solution of the two dimensional RLW equation of are obtained by elliptic integral method, and some estimations the uniqueness and the stability of the periodic solution with both x, y to the Cauchy problem are proved by the priori estimations.
Abstract: A cnoidal wave solution of the two dimensional RLW equation of are obtained by elliptic integral method, and the some estimations the uniqueness and the stability of the periodic solution with both x, y to the Cauchy problem are proved by the priori estimations.
TL;DR: In this paper, the authors present a Maple packaged automated Jacobi elliptic function method, which can entirely automatically output the exact periodic wave solutions to nonlinear evolution equations, and demonstrate the effectiveness of this method using as examples the application to a variety of equations with physical interest.
Abstract: We describe the Jacobi elliptic function method for finding exact periodic wave solutions to nonlinear evolution equations. We present a Maple packaged automated Jacobi elliptic function method, which can entirely automatically output the exact periodic wave solutions. The effectiveness of the automated Jacobi elliptic function method is demonstrated using as examples the application to a variety of equations with physical interest. Not only are the previously known solutions recovered but in some cases new solutions and more general forms of solutions are obtained.
TL;DR: The existence of dispersion-managed solitons for the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is well known in nonlinear optics.
Abstract: The existence of “dispersion-managed solitons,” ie, stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models Hence, here we consider as a basic model the variable-coefficient Korteweg–de Vries equation; this has the form of a Korteweg–de Vries equation with a periodically varying third-order dispersion coefficient, that can take both positive and negative values More generally, this model may be extended to include fifth-order dispersion Such models may describe, for instance, periodically modulated waveguides for long gravity-capillary waves We develop an analytical approximation for solitary waves in the weakly nonlinear case, from which it is possible to obtain a reduction to a relatively simple integral equation, which is readily solved numerically Then, we describe some systematic direct simulations of the full equation, which use the soliton shape produced by the integral equation as an initial condition These simulations reveal regions of stable and unstable pulsating solitary waves in the corresponding parametric space Finally, we consider the effects of fifth-order dispersion
TL;DR: The conditions for the capture or reflection of a solitary wave by a single localized external force are obtained, with an emphasis on the role of the cubic nonlinear term.
Abstract: The interaction of a strongly nonlinear solitary wave with an external force is studied using the extended Korteweg–de Vries equation as a model. This equation has several different families of nonlinear wave solutions: solitons, the so-called "thick" solitons, algebraic solitons and breathers, depending upon the sign of the cubic nonlinear term. A simple nonlinear dynamical system of the second order for the amplitude and position of the solitary wave is derived, and used to study the interaction. Its solutions are investigated in the phase plane. The conditions for the capture or reflection of a solitary wave by a single localized external force are obtained, with an emphasis on the role of the cubic nonlinear term.
TL;DR: In this article, an extended Benjamin-Bona-Mahony (eBBM) equation is used to model the solitary wave interaction and the mass and energy of the dispersive wavetrain generated by the inelastic collision.
Abstract: Solitary wave interaction is examined using an extended Benjamin-Bona-Mahony (eBBM) equation. This equation includes higher-order nonlinear and dispersive effects and is is asymptotically equivalent to the extended Korteweg-de Vries (eKdV) equation. The eBBM formulation is preferable to the eKdV equation for the numerical modelling of solitary wave collisions, due to the stability of its finite-difference scheme. In particular, it allows the interaction of steep waves to be modelled, which due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of solitary wave collisions of varying nonlinearity are performed for two special cases corresponding to surface water waves. The mass and energy of the dispersive wavetrain generated by the inelastic collision is tabulated and used to show that the change in solitary wave amplitude after interaction is of , verifying previously obtained theoretical predictions.
TL;DR: In this paper, an extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs), taking full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein-Gordon equation.
Abstract: An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein-Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.
01 Jan 2002
TL;DR: In this paper, a cnoidal wave solution of the two dimensional RLW equation was obtained by elliptic integral method, and the uniqueuess and the stability of the periodicsolution with both x, y to the Cauchy problem were proved by the priori estimations.
Abstract: A cnoidal wave solution of the two dimensional RLW equation of are obtained by elliptic integral method, and the somw estimations the uniqueuess and the stability of the periodicsolution with both x, y to the Cauchy problem are proved by the priori estimations .
TL;DR: In this article, the wave packet theory was used to obtain all the solutions of the weakly damped nonlinear Schrodinger equation and the bifurcation phenomenon exists in both steady and non-steady solutions.
Abstract: Using the wave packet theory, we obtain all the solutions of the weakly damped nonlinear Schrodinger equation. These solutions are the static solution and solutions of planar wave, solitary wave, shock wave and elliptic function wave and chaos. The bifurcation phenomenon exists in both steady and non-steady solutions. The chaotic and periodic motions can coexist in a certain parametric space region.
TL;DR: In this paper, the wave propagation from the Deep Ocean to a shoreline has been numerically modeled and model equations govern combined effects of shoaling, refraction, diffraction and breaking.
Abstract: Propagation of waves from Deep Ocean to a shoreline has been numerically modeled. Model equations govern combined effects of shoaling, refraction, diffraction and breaking. Linear, harmonic, and irrotational waves are considered, and the effects of currents and reflection on the wave propagation are assumed to be negligible. To describe the wave motion, mild slope equation has been decomposed into three equations that are solved in terms of wave height, wave approach angle and wave phase function. It is assumed that energy propagates along the wave crests, however, the wave phase function changes to handle any horizontal variation in the wave height. Model does not have the limitation that one coordinate should follow the dominant wave direction. Different wave approach angles can be investigated on the same computational grid. Finite difference approximations have been applied in the solution of governing equations. Model predictions are compared with the results of semicircular shoal tests perf...