Showing papers on "Cnoidal wave published in 2015"
TL;DR: In this paper, a consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a CRE equation, and a system having a CRE is then defined to be CRE solvable.
Abstract: A consistent Riccati expansion (CRE) is proposed for solving nonlinear systems with the help of a Riccati equation. A system having a CRE is then defined to be CRE solvable. The CRE solvability is demonstrated quite universal for various integrable systems including the Korteweg–de Vries, Kadomtsev–Petviashvili, Ablowitz–Kaup–Newell–Segur (and then nonlinear Schrodinger), sine-Gordon, Sawada–Kotera, Kaup–Kupershmidt, modified asymmetric Nizhnik–Novikov–Veselov, Broer–Kaup, dispersive water wave, and Burgers systems. In addition, it is revealed that many CRE solvable systems share a similar determining equation describing the interactions between a soliton and a cnoidal wave. They have a common nonlocal symmetry expression and they possess a formally universal once Backlund transformation.
142 citations
26 Mar 2015
TL;DR: In this paper, the authors derive conservation laws for solutions of wave equations and derive Strichartz estimates to deal with semi-linear problems with data (f, g) ∈ Hs × Hs−1 for large enough s.
Abstract: where := −∂2 t +∆ and u[0] := (u, ut)|t=0. The equation is semi-linear if F is a function only of u, (i.e. F = F (u)), and quasi-linear if F is also a function of the derivatives of u (i.e. F = F (u,Du), where D := (∂t,∇)). The goal is to use energy methods to prove local well-posedness for quasilinear equations with data (f, g) ∈ Hs × Hs−1 for large enough s, and then to derive Strichartz estimates to deal with semi-linear problems with data (f, g) ∈ Ḣ1 × L2. We begin, however, by deriving various conservation laws for solutions of wave equations.
85 citations
TL;DR: In this article, a nonlinear model for the flow of an incompressible and inviscid fluid given in Part I, the wave-induced loads on the submerged, fixed (and rigid) plate are calculated, and results are compared with the available laboratory data, and with linear solutions of the problem.
62 citations
TL;DR: In this paper, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation.
Abstract: Abstract In this article, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. With help of Maple and by using the multiple exp-method, we can get exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions. Furthermore, we apply the linear superposition principle to find n-wave solutions of the CBS equation. Two cases with specific values of the involved parameters are plotted for each two-wave and three-wave solutions.
51 citations
TL;DR: Using the Painleve test, it is shown that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform, but there are some exact solutions of the generalized Dries equation for two forms of the source.
50 citations
TL;DR: It is found that the features (amplitude and width) of nonlinear ion-acoustic cnoidal waves are proportional to plasma number density, ion cyclotron frequency, and direction cosines.
Abstract: The complex pattern and propagation characteristics of nonlinear periodic ion-acoustic waves, namely, ion-acoustic cnoidal waves, in a dense relativistic degenerate magnetoplasma consisting of relativistic degenerate electrons and nondegenerate cold ions are investigated. By means of the reductive perturbation method and appropriate boundary conditions for nonlinear periodic waves, a nonlinear modified Korteweg--de Vries (KdV) equation is derived and its cnoidal wave is analyzed. The various solutions of nonlinear ion-acoustic cnoidal and solitary waves are presented numerically with the Sagdeev potential approach. The analytical solution and numerical simulation of nonlinear ion-acoustic cnoidal waves of the nonlinear modified KdV equation are studied. Clearly, it is found that the features (amplitude and width) of nonlinear ion-acoustic cnoidal waves are proportional to plasma number density, ion cyclotron frequency, and direction cosines. The numerical results are applied to high density astrophysical situations, such as in superdense white dwarfs. This research will be helpful in understanding the properties of compact astrophysical objects containing cold ions with relativistic degenerate electrons.
50 citations
01 Feb 2015
TL;DR: In this paper, the horizontal and vertical wave forces due to the interaction of cnoidal waves with a two-dimensional, horizontal flat plate located in shallow-water are studied through laboratory experiments and calculations.
Abstract: Horizontal and vertical wave forces due to the interaction of cnoidal waves with a two-dimensional, horizontal flat plate located in shallow-water are studied through laboratory experiments and calculations. The experiments are conducted for a combination of two water depths, five wavelengths and four wave heights, corresponding to the propagation of nonlinear waves in shallow-water depth. The model is located at six different elevations and submergence depths such that all possible cases of a coastal bridge deck fully above the still-water level, a deck on the surface and a fully submerged deck are considered in the study. Calculations are performed for the same cases as in the laboratory experiments and include the results of a nonlinear shallow-water wave model based on the Level I Green–Naghdi equations for the fully submerged cases, and Euler’s equations coupled with the Volume of Fluid interface tracking method for one submerged case, one elevated case, and one case at the water surface. Comparison of existing theoretical solutions are also provided, including the Long-Wave Approximation based on linear potential theory for the submerged cases, and empirical relations for the elevated cases. The set of data presented here provides an insight into storm wave loads on the decks of coastal bridges, jetties and piers located in shallow-water areas.
48 citations
TL;DR: In this paper, it was shown that Cnoidal waves constitute fundamental material instabilities stemming from the propagation of elasto-plastic P-waves, and they are the singularities of the problem of homogeneous volumetric deformation of a rate-dependent, heterogeneous solid material.
Abstract: Cnoidal waves are nonlinear and exact periodic stationary waves, well known in the shallow water theory of fluid mechanics. In this study we retrieve such periodic stationary wave solutions as singularities of the problem of homogeneous volumetric deformation of a rate-dependent, heterogeneous solid material. In accordance to the classical Hill stationary wave localization instability, which provides velocity gradient discontinuities in shear failure, cnoidal waves are dilational and compactional manifestations of volumetric localization along lines of stress discontinuities. They therefore emerge along the volumetric component of the classical slip line field theory, with their regular distance being a tell tale indication of rate-dependent volumetric deformation. We discuss applications for the dominant mode of I1 compaction in geomaterials where distinct cnoidal wave instabilities appear as localisation features in compaction. We also discuss the case of localisation features in a classical (J2 plastic) material where a small but important cnoidal contribution may trigger equidistant bands of localisation known as Luders lines. We therefore postulate that cnoidal waves constitute fundamental material instabilities stemming from the propagation of elasto-plastic P-waves.
46 citations
TL;DR: In this article, the horizontal and vertical forces on a 1:35 scale model of a typical two-lane coastal bridge due to cnoidal wave loads are investigated by conducting an extensive set of laboratory experiments and comparing the resulting data with CFD calculations and existing simplified design-type equations.
Abstract: Horizontal and vertical forces on a 1:35 scale model of a typical two-lane coastal bridge due to cnoidal wave loads are investigated by conducting an extensive set of laboratory experiments and comparing the resulting data with CFD calculations and existing simplified, design-type equations. The experimental parameters tested cover a wide range of wave and inundation conditions that may occur during a major storm or hurricane. This includes a wave matrix of 40 waves and bridge model elevations covering a range where the top of the bridge is fully submerged below the still-water level (SWL) to where the bottom of the girders are elevated above the SWL. Measurements for surface elevation, vertical and horizontal forces are compared with calculations made by solving Euler’s equations using the CFD software OpenFOAM with good agreement. Vertical uplift and horizontal positive forces (forces measured in the direction of wave propagation) are compared with the simplified equations using the relations given in Douglass et al. (2006). This set of data provides a valuable benchmark for understanding wave loads on coastal bridges during a storm or hurricane.
38 citations
TL;DR: In this article, the exact solutions for the fractional Korteweg-de Vries equations and the coupled Kortwé-deVries equations with time-fractional derivatives using the functional variable method using the modified Riemann-Liouville derivative sense were presented.
Abstract: This paper presents the exact solutions for the fractional Korteweg–de Vries equations and the coupled Korteweg–de Vries equations with time-fractional derivatives using the functional variable method The fractional derivatives are described in the modified Riemann–Liouville derivative sense It is demonstrated that the calculations involved in the functional variable method are extremely simple and straightforward and this method is very effective for handling nonlinear fractional equations
34 citations
TL;DR: In this article, the authors used the bifurcation method of dynamical systems to investigate the nonlinear wave solutions of the modified Benjamin-Bona-Mahony equation.
Abstract: In this paper, we use the bifurcation method of dynamical systems to investigate the nonlinear wave solutions of the modified Benjamin–Bona–Mahony equation. These nonlinear wave solutions contain periodic wave solutions, solitary wave solutions, periodic blow-up wave solutions, kink wave solutions, unbounded wave solutions and blow-up wave solutions. Some previous results are extended.
TL;DR: In this paper, a new extension of the (G′/G)-expansion method for finding the solitary wave solutions of the modified Korteweg-de Vries (mKdV) equation was proposed.
Abstract: In this present work, we have studied new extension of the (G′/G)-expansion method for finding the solitary wave solutions of the modified Korteweg–de Vries (mKdV) equation. It has been shown that the proposed method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. The obtained results show that the method is very powerful and convenient mathematical tool for nonlinear evolution equations in science and engineering.
TL;DR: The key point is to relate λ(a) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.
TL;DR: In this paper, a chain of infinitely many particles coupled by nonlinear springs obeying the equations of motion 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 $$\begin{aligned} \ddot{q}_n = V'(q_{n+1}-q_n) - V'(q_n-q_{n-1}) \end{aligned}$$
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 (Nonlinearity 12:1601–1627, 1999) to a periodic setting and the spectral theory of the periodic Schrodinger operator with KdV cnoidal wave potential.
TL;DR: It is found here that a nonlinear splitting of the wave-number spectrum at the zero-dispersion point, where energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation, is responsible for theWave-packet formation.
Abstract: The long-time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. There is currently no entirely satisfactory explanation as to why these wave packets form. Here the initial value problem is considered within the context of the Gardner-Ostrovsky, or rotation-modified extended Korteweg-de Vries, equation. The linear Gardner-Ostrovsky equation has maximum group velocity at a critical wave number, often called the zero-dispersion point. It is found here that a nonlinear splitting of the wave-number spectrum at the zero-dispersion point, where energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation, is responsible for the wave-packet formation. Numerical comparisons of the decay of a solitary wave in the Gardner-Ostrovsky equation and a derived nonlinear Schrodinger equation at the zero-dispersion point are used to confirm the spectral splitting.
TL;DR: In this paper, the truncated Painleve analysis and the consistent tanh expansion (CTE) method are developed for the (2+1)-dimensional breaking soliton equation.
Abstract: In this paper, the truncated Painleve analysis and the consistent tanh expansion (CTE) method are developed for the (2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.
TL;DR: In this article, the authors investigated neutral and buoyant particle motions in an irrotational flow under the passage of linear, nonlinear gravity and weakly nonlinear solitary waves at a constant water depth.
Abstract: Neutral and buoyant particle motions in an irrotational flow are investigated under the passage of linear, nonlinear gravity, and weakly nonlinear solitary waves at a constant water depth. The developed numerical models for the particle trajectories in a non-turbulent flow incorporate particle momentum, size, and mass (i.e., inertial particles) under the influence of various surface waves such as Korteweg-de Vries waves which admit a three parameter family of periodic cnoidal wave solutions. We then formulate expressions of mass-transport velocities for the neutral and buoyant particles. A series of test cases suggests that the inertial particles possess a combined horizontal and vertical drifts from the locations of their release, with a fall velocity as a function of particle material properties, ambient flow, and wave parameters. The estimated solutions exhibit good agreement with previously explained particle behavior beneath progressive surface gravity waves. We further investigate the response of a neutrally buoyant water parcel trajectories in a rotating fluid when subjected to a series of wind and wave events. The results confirm the importance of the wave-induced Coriolis-Stokes force effect in both amplifying (destroying) the pre-existing inertial oscillations and in modulating the direction of the flow particles. Although this work has mainly focused on wave-current-particle interaction in the absence of turbulence stochastic forcing effects, the exercise of the suggested numerical models provides additional insights into the mechanisms of wave effects on the passive trajectories for both living and nonliving particles such as swimming trajectories of plankton in non-turbulent flows.
TL;DR: In this article, the interaction of successive solitary waves in the swash zone has been studied using large-scale experiments with a simple bathymetry of a constant depth region, where the water depth was 1.72m, and a plane beach, whose slope was 1:12.
TL;DR: In this article, an integrated analytical-numerical model to simulate a solitary wave propagating past a fixed and partially immersed body is presented. And the velocity potentials of the inner fluid region beneath the structure are determined analytically with unknown coefficients evaluated from a system of newly formulated matching equations with the uses of continuous velocity and velocity potential as well as the orthogonal properties of the eigenfunctions.
TL;DR: The (1+1)-dimensional nonlinear Klein-Gordon-Zakharov equation is considered as a model equation for describing the interaction of the Langmuir wave and the ion acoustic wave in high frequency plasma.
TL;DR: In this paper, a general depth-averaged model based on the k-ϵ turbulence closure was developed for undular hydraulic jumps, in which both streamline curvature and vorticity are accounted for.
Abstract: An undular hydraulic jump corresponds to the weak transition from super- to subcritical-flow in the form of steady free surface undulations. Previous models on undular hydraulic jumps employed the potential flow theory, i.e., the solitary and cnoidal wave theories. Experimental observations indicate the inadequacy of this theory, which motivated the development of more advanced approximations. Basic flow features including friction effects on the velocity profile, modelling of the bed-shear stress, and Reynolds stresses are considered. However, none of the models currently available include all these aspects. In this study, a general depth-averaged model is developed based on the k-ϵ turbulence closure. The general depth-averaged equations are applied to the undular jump problem, introducing a suitable time-averaged velocity distribution based on a composite power-law model, in which both streamline curvature and vorticity are accounted for. The bed-shear stress closure is included by a boundary l...
TL;DR: In this paper, the exact traveling wave solutions of the fifth-order Kaup-Kuperschmidt equation were investigated and the relationship of the two subequations and the two known rst integrals were analyzed.
Abstract: In this paper we investigate the exact traveling wave solutions of the fifth-order Kaup-Kuperschmidt equation. The bifurcation and exact solutions of a general first-order nonlinear equation are investigated firstly. With the help of Maple and by using the bifurcation and exact solutions of two derived subequations, we obtain two families of solitary wave solutions and two families of periodic wave solutions of the KK equation. The relationship of the two subequations and the two known rst integrals are analyzed.
TL;DR: In this paper, a generalization of the Korteweg-de-vries equation obtained from the Fermi-Pasta-Ulam problem is studied and the necessary condition for existence of the meromorphic solution is carried out and some exact solutions can be found.
TL;DR: The statistical properties of long-crested nonlinear wave time series measured in an offshore basin have been analyzed in different aspects such as the distributions of surface elevation, wave crest, wave trough, and wave period as discussed by the authors.
TL;DR: The dynamical behaviors of (1 + 1)-dimensional analytical nonautonomous cnoidal waves and solitons in the exponential and hyperbolic diffraction decreasing waveguides with the self-focusing and self-defocusing nonlinearities are studied, respectively.
TL;DR: It is proved that in the presence of dispersion effect on the equation, yet there exists bounded traveling wave solutions in different classes in terms of solitary waves, periodic and elliptic functions in certain regions.
TL;DR: In this paper, a cnoidal wave-seabed-pipeline system is modeled using the finite element method and the precise integration method is used to estimate the Jacobian elliptic function.
TL;DR: In this paper, the authors derived new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation by using a method which they introduced here for the first time.
Abstract: In this paper we derive new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation by using a method which we introduce here for the first time. Firstly, we reduce the associated fourth-order nonlinear ordinary differential equation (ODE) into a solvable first-order nonlinear ODE to obtain new exact traveling wave solutions, including the solitary wave and periodic solutions. Furthermore, using the new method we derive the quasi-periodic wave solutions of this equation by assuming that the solutions of the corresponding higher-order ODE are the sum of the solutions of two solvable first-order nonlinear ODEs. This new method can be used to investigate the exact traveling wave solutions and quasi-periodic wave solutions of a general class of higher-order wave equations.
TL;DR: A new model equation describing weakly nonlinear long internal waves at the interface between two thin layers of different density is derived for the specific relationships between the densities, layer thicknesses and surface tension between the layers.
TL;DR: In this article, the linear propagation of an electron wave in a plasma whose distribution function, at zero order in the wave amplitude, may be chosen arbitrarily, provided that it is not strongly peaked at the wave phase velocity, and that it varies very little over one wave period and one wavelength.
Abstract: This paper addresses the linear propagation of an electron wave in a plasma whose distribution function, at zero order in the wave amplitude, may be chosen arbitrarily, provided that it is not strongly peaked at the wave phase velocity, and that it varies very little over one wave period and one wavelength. Then, from first principles is derived an equation for the wave action density that allows for Landau damping, whose rate is calculated at first order in the variations of the wave number and frequency. Moreover, the effect of collisions is accounted for in a way that adapts to any choice for the collision operator in Boltzmann equation. The wave may also be externally driven, so that the results presented here apply to stimulated Raman scattering.