Showing papers on "Cnoidal wave published in 2012"
01 Jan 2012
TL;DR: In particular, the linear dispersive term in the Korteweg-de Vries equation prevents this from ever happening in its solution as discussed by the authors, and the instability and subsequent modulation of an initially uniform wave profile can be prevented by including dispersive effects in the shallow water theory.
Abstract: Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.
864 citations
TL;DR: In this article, the travelling wave solutions involving parameters of the (2 + 1)-dimensional dispersive long wave equation were constructed by using a new approach, namely, the −expansion method.
52 citations
TL;DR: Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, a new elliptic function equation was used to get a new kind of solutions of nonlinear evolution equations.
Abstract: Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (1+1)-dimensional and higher dimensional systems.
46 citations
TL;DR: In this article, the Korteweg-de Vries equation was derived for ion acoustic nonlinear periodic (cnoidal) wave and soliton solutions in unmagnetized pair-ion plasmas consisting of the same mass ion species with different temperatures.
Abstract: Electrostatic acoustic nonlinear periodic (cnoidal) waves and solitons are investigated in unmagnetized pair-ion plasmas consisting of the same mass ion species with different temperatures. It is found that the temperature difference between negatively and positively charged ions appropriates the dispersion property to linear acoustic waves and this difference has a decisive role in nonlinear dynamics as well. Using a reductive perturbation method and appropriate boundary conditions the Korteweg–de Vries equation is derived. Both cnoidal wave and soliton solutions are discussed in detail. In the special case, it is revealed that the amplitude of a soliton may become larger than what is allowed by the nonlinear stationary wave theory, which is equal to the quantum tunneling by a particle through a potential barrier effect. The serious flaw in the results obtained for ion acoustic nonlinear periodic waves by Yadav et al (1995 Phys. Rev. E 52 3045) in two-electron temperature plasmas and Chawla and Misra (2010 Phys. Plasmas 17 102315) in electron–positron–ion plasmas is also pointed out.
37 citations
TL;DR: In this paper, the authors used the -expansion method to construct travelling wave solutions of nonlinear evolution equations, expressed in terms of hyperbolic functions, trigonometric functions and rational functions.
Abstract: In this work, we establish exact solutions for (2 + 1)- and (3 + 1)-dimensional shallow water wave equations. The -expansion method is used to construct travelling wave solutions of nonlinear evolution equations. The travelling wave solutions are expressed in terms of hyperbolic functions, trigonometric functions and rational functions. This method presents a wider applicability for handling nonlinear wave equations.
34 citations
TL;DR: In this paper, a long wave scaling for the Vlasov-Poisson equation and the Korteweg-De Vries equation in the presence of an external magnetic field was introduced.
Abstract: We introduce a long wave scaling for the Vlasov-Poisson equation and derive, in the cold ions limit, the Korteweg-De Vries equation (in 1D) and the Zakharov-Kuznetsov equation (in higher dimensions, in the presence of an external magnetic field). The proofs are based on the relative entropy method.
33 citations
TL;DR: In this article, the method recently developed by Demina and Kudryashov is applied to the Olver water wave equation and new solutions of this equation are found in terms of the Weierstrass elliptic function.
30 citations
29 citations
TL;DR: Choosing some special parameters, the method of dynamical systems is applied to a generalized two-component Camassa–Holm system and exact parametric representations of the traveling wave solutions are obtained.
Abstract: In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.
27 citations
TL;DR: In this paper, a phase-resolving nonlinear frequency-domain model with both wave-current interaction and viscous mud-induced energy dissipation is discussed, and the model is compared to dissipation rates deduced from experimental data, with favorable results.
17 citations
TL;DR: In this paper, the periodic and solitary wave solutions of two selected systems of nonlinear wave equations are obtained using the first integral method, and this approach can also be applied to other coupled nonlinear differential equations.
Abstract: In this paper, the periodic and solitary wave solutions of two selected systems of nonlinear wave equations are obtained using the first integral method. The long?short-wave interaction system and Bogoyavlenskii equations?are considered as examples, and this approach can also be applied to other coupled nonlinear differential equations.
TL;DR: In this paper, coupled evolution equations for ion-acoustic waves in a collisionless, unmagnetized plasma consisting of hot isothermal electrons, cold ions, and massive mobile charged dust grains are derived.
Abstract: Using reductive perturbation method with appropriate boundary conditions, coupled evolution equations for first and second order potentials are derived for ion-acoustic waves in a collisionless, un-magnetized plasma consisting of hot isothermal electrons, cold ions, and massive mobile charged dust grains. The boundary conditions give rise to renormalization term, which enable us to eliminate secular contribution in higher order terms. Determining the non secular solution of these coupled equations, expressions for wave phase velocity and averaged non-linear ion flux associated with ion-acoustic cnoidal wave are obtained. Variation of the wave phase velocity and averaged non-linear ion flux as a function of modulus (k2) dependent wave amplitude are numerically examined for different values of dust concentration, charge on dust grains, and mass ratio of dust grains with plasma ions. It is found that for a given amplitude, the presence of positively (negatively) charged dust grains in plasma decreases (increases) the wave phase velocity. This behavior is more pronounced with increase in dust concentrations or increase in charge on dust grains or decrease in mass ratio of dust grains. The averaged non-linear ion flux associated with wave is positive (negative) for negatively (positively) charged dust grains in the plasma and increases (decreases) with modulus (k2) dependent wave amplitude. For given amplitude, it increases (decreases) as dust concentration or charge of negatively (positively) charged dust grains increases in the plasma.
01 Jan 2012
TL;DR: In this article, the Jacobi doubly periodic wave solutions for the Benjamin-Bona-Mahony equation and the Korteweg-de Vries equation were constructed by applying the extended Jacobi elliptic function expansion method.
Abstract: The Benjamin-Bona-Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg-de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. Using a computerized symbolic computation technique, we construct the interesting Jacobi doubly periodic wave solutions for these equations by applying the extended Jacobi elliptic function expansion method.
TL;DR: In this article, the Jacobi elliptic function method with symbolic computation is extended to these nonlinear equations for constructing their doubly periodic wave solutions, leading to new Jacobi doubly-periodic wave solutions and soliton solutions.
Abstract: Arrays of vortices are considered for two-dimensional inviscid flows when the functional relationship between the stream function and the vorticity is hyperbolic sine, exponential, sine, and power functions. The Jacobi elliptic function method with symbolic computation is extended to these nonlinear equations for constructing their doubly periodic wave solutions. The different Jacobi function expansions may lead to new Jacobi doubly periodic wave solutions, triangular periodic solutions and soliton solutions. In addition, as an illustrative sample, the properties for the Jacobi doubly periodic wave solutions of the nonlinear equations are shown with some figures.
TL;DR: In this article, the authors investigated the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2 + 1) dimensional Davey-Stewartson (DS) equation governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice.
Abstract: This paper investigates the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2+1) dimensional Davey-Stewartson (DS) equation which is governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice. By using extended mapping method technique, we have shown that the 2DRLG-2DDS equations can be reduced to the elliptic-like equation. Then, the extended mapping method is used to obtain a series of solutions including the single and the combined non degenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear partial differential equations (NLPDEs).
TL;DR: In this article, the oblique propagation of nonlinear periodic ion-acoustic waves in magnetized dusty plasma is investigated, and the equations describing the dynamics of the wave potential in the first and second orders of the perturbation theory are derived, and their nonsecular periodic solutions are found.
Abstract: The oblique propagation of nonlinear periodic ion-acoustic waves in magnetized dusty plasma is investigated. The equations describing the dynamics of the wave potential in the first and second orders of the perturbation theory are derived, and their nonsecular periodic solutions are found. The average nonlinear ion flux caused by the propagation of a cnoidal wave is estimated. The magnitude and direction of the ion flux are analyzed as functions of the dust charge density and the angle between the wave propagation direction and the magnetic field.
TL;DR: In this article, the existence and stability of standing waves for the periodic cubic nonlinear Schrodinger equation with a point defect determined by the periodic Dirac distribution at the origin was studied.
Abstract: We study the existence and stability of standing waves for the periodic cubic nonlinear Schrodinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic-peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal-peak standing wave solutions are unstable in with respect to perturbations which have the same period as the wave itself. Global well-posedness is verified for the Cauchy problem in .
TL;DR: In this article, the wave attenuation properties of the double trapezoidal submerged breakwaters on the flat-bed were investigated by conducting physical experiments subjected to linear and cnoidal incident waves.
Abstract: This paper investigates the wave attenuation properties of the double trapezoidal submerged breakwaters on the flat-bed by conducting physical experiments subjected to linear and cnoidal incident waves. The method of Goda’s two points is used to separate the heights of incident, reflected and transmitted waves based on the experimental data. The possible factors affecting the wave attenuation properties of the double trapezoidal submerged breakwaters (i.e., the relative submerged water depth, relative breakwater spacing, wave steepness and relative wave height) are investigated with respect to the reflection and transmission coefficients. The results show that there is a range, within which the breakwater spacing has little impact on the reflection coefficient, and the transmission coefficient tends to be a constant. The influence of the wave steepness is reduced while the breakwater spacing is too large or too small. Within the range of the relative wave height tested in this study, the reflection and transmission coefficients increase and decrease with the relative wave height, respectively. The double trapezoidal submerged breakwaters model indicates a good attenuation effect for larger wave steepness, big relative wave height and within the range of the relative breakwater spacing between 12.5 and 14 according to linear and cnoidal waves. The changes of wave energy spectra between the double submerged breakwaters on the flat-bed are investigated by the fast Fourier transform (FFT) method, showing that wave energy dissipation can be reached more effectively when the relative breakwater spacing is 12.5.
TL;DR: In this paper, the effect of mutual rectification of two electromagnetic waves in a graphene superlattice is studied for a cnoidal wave polarized along the axis and a sinusoidal wave polarization along the orthogonal direction, and an expression for the density of the direct current generated by mixing of these waves is obtained.
Abstract: The effect of mutual rectification of two electromagnetic waves in a graphene superlattice is studied for a cnoidal wave polarized along the superlattice axis and a sinusoidal wave polarized along the orthogonal direction. An expression for the density of the direct current generated by mixing of these waves is obtained. The direct current is studied as a function of the cnoidal wave amplitude. The possibility of varying both the value and the direction of the direct current with varying the cnoidal wave amplitude is shown.
TL;DR: In this paper, a direct symbolic computation method combined with variable transformations is applied to the short-pulse equation in nonlinear optics, and some new Jacobi elliptic function solutions are obtained.
Abstract: Applying a direct symbolic computation method combined with variable transformations, some new Jacobi elliptic function solutions are obtained to the short-pulse equation in nonlinear optics. When Jacobi elliptic function modulus m→1 or 0, the travelling wave solutions degenerate to two types of solutions, namely, the loop-like soliton solution and the trigonometric function solution.
TL;DR: In this article, the authors developed a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs, based on a Korteweg-de Vries (KdV) equation for the perturbation wavenumber on a periodic background.
Abstract: A dark solitary wave, in one space dimension and time, is a wave that is bi-asymptotic to a periodic state, with a phase shift, and with localized modulation in between. The most well-known case of dark solitary waves is the exact solution of the defocusing nonlinear Schrodinger equation. In this paper, our interest is in developing a mechanism for the emergence of dark solitary waves in general, and not necessarily integrable, Hamiltonian PDEs. The focus is on the periodic state at infinity as the generator. It is shown that a natural mechanism for the emergence is a transition between one periodic state that is (spatially) elliptic and another one that is (spatially) hyperbolic. It is shown that the emergence is governed by a Korteweg–de Vries (KdV) equation for the perturbation wavenumber on a periodic background. A novelty in the result is that the three coefficients in the KdV equation are determined by the Krein signature of the elliptic periodic orbit, the curvature of the wave action flux and the slope of the wave action, with the last two evaluated at the critical periodic state.
TL;DR: In this article, a method for calculating the Cnoidal function in cnoidal wave theory is presented, which is based on the precise integration method and an improvement on the data preliminary treatment.
TL;DR: In this paper, the effective dynamics of interacting waves for coupled Schrodinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied.
Abstract: The effective dynamics of interacting waves for coupled Schrodinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large amplitude and a long wavelength, the dynamics of the surface of the fluid is described by the Schrodinger equation, while that of the internal wave is described by the Korteweg-de Vries equation. The purpose of this letter is to show that in the presence of a slowly varying random bottom, the coupled waves evolve adiabatically over a long time scale. The analysis covers the cases when the surface wave is a stable bound state or a long-lived metastable state.
TL;DR: It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.
Abstract: An integrable 2-component Camassa-Holm (2-CH) shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.
TL;DR: Using an auxiliary ordinary differential equation method, many exact travelling wave solutions are obtained for the (2 + 1) dimensional Nizhnik–Novikov–Veselov equation which is considered as a model for an incompressible fluid.
TL;DR: A variety of exact solutions for the Kadomtsov-Petviashvili- Benjamin-Bona-Mahony (KP-BBM) equation, nonlinear Zakharov-Kuznetsov- Benjamin, Bona- Mahony (ZK-B BM) equation and the generalized ZK- BBM equation are developed by means of the extended Jacobi elliptic function expansion method.
Abstract: A variety of exact solutions for the Kadomtsov-Petviashvili- Benjamin-Bona-Mahony (KP-BBM) equation, nonlinear Zakharov-Kuznetsov- Benjamin-Bona-Mahony (ZK-BBM) equation and the generalized ZK-BBM equation are developed by means of the extended Jacobi elliptic function expansion method. Soliton and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions.
Key words: Jacobi elliptic function method, soliton and triangular periodic solutions, nonlinear dispersive Kadomtsov-Petviashvili- Benjamin-Bona-Mahony (KP-BBM) equation, nonlinear Zakharov-Kuznetsov- Benjamin-Bona-Mahony (ZK-BBM) equation, the generalized ZK-BBM equation.
TL;DR: A new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi ellic functions, are very effective to uniformly construct more new exact periodic solutions.
Abstract: With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.
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TL;DR: A vector solution to the spherical wave equation is presented in this article, which requires rotation of the wave media and a minor revision to the form of the Wave Equation (WE).
Abstract: A vector solution to the spherical wave equation is presented. The solution requires rotation of the wave media and a minor revision to the form of the wave equation.
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TL;DR: In this paper, a chain of infinitely many particles coupled by nonlinear springs obeying the equations of motion with generic nearest-neighbour potential was studied, and it was shown that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves.
Abstract: We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1})] with generic nearest-neighbour potential $V$. We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength$^{-3}$ suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of Friesecke and Pego (1999) to a periodic setting and the spectral theory of the periodic Schr\"odinger operator with KdV cnoidal wave potential.
01 Feb 2012
TL;DR: In this paper, the coupled dispersive (2+1)-dimensional long wave equation is studied with power law nonlinearity where the ansatz method is applied to yield solitary wave solutions.
Abstract: In this paper, the coupled dispersive (2+1)-dimensional long wave equation is studied. The traveling wave hypothesis yields complexiton solutions. Subsequently, the wave equation is studied with power law nonlinearity where the ansatz method is applied to yield solitary wave solutions. The constraint conditions for the existence of solitons naturally fall out of the derivation of the soliton solution.