Showing papers on "Cnoidal wave published in 2009"
TL;DR: In this paper, the traveling wave solutions involving parameters of the combined Korteweg-de Vries modified KORTeweg de Vries equation, reaction-diffusion equation, compound KdV-Burgers equation, and generalized shallow water wave equation were constructed using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation.
Abstract: I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.
270 citations
TL;DR: In this paper, the Hirota bilinear method is extended to explicitly construct multi-periodic wave solutions for the asymmetrical Nizhnik-Novikov-Veselov equation.
Abstract: Based on a multi-dimensional Riemann theta function, the Hirota bilinear method is extended to explicitly construct multi-periodic (quasi-periodic) wave solutions for the asymmetrical Nizhnik–Novikov–Veselov equation. Among these periodic waves, two-periodic waves are a direct generalization of well-known cnoidal waves; their surface pattern is two dimensional. The main physical result is the description of the behavior of nonlinear waves in shallow water. A limiting procedure is presented to analyze asymptotic properties of the two-periodic waves in details. Relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a 'small amplitude' limit.
105 citations
TL;DR: In this paper, an exact solitary wave solution of the Korteweg-de Vries equation with power law nonlinearity with time-dependent coefficients of the nonlinear as well as the dispersion terms was obtained.
Abstract: This paper obtains an exact solitary wave solution of the Korteweg–de Vries equation with power law nonlinearity with time-dependent coefficients of the nonlinear as well as the dispersion terms. In addition, there are time-dependent damping and dispersion terms. The solitary wave ansatz is used to carry out the analysis. It is only necessary for the time-dependent coefficients to be Riemann integrable. As an example, the solution of the special case of cylindrical KdV equation falls out.
90 citations
TL;DR: The first integral method was used to construct travelling wave solutions of the Cahn–Allen equation and the obtained results include periodic and solitary wave solutions.
72 citations
TL;DR: In this article, the authors studied the orbital stability of a family of periodic stationary traveling wave solutions to the generalized Korteweg-de Vries equation and derived sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ODE.
Abstract: In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg–de Vries equation $u_t=u_{xxx}+f(u)_x$. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equation restricted to the manifold of periodic traveling wave solutions. We show this condition is equivalent to the solution being spectrally stable with respect to perturbations of the same period in the case when $f(u)=u^2$ (the Korteweg–de Vries equation) and in neighborhoods of the homoclinic and equilibrium solutions if $f(u)=u^{p+1}$ for some $p\geq1$.
70 citations
TL;DR: In this paper, it was shown that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds, each of which are solutions to a Korteweg-de Vries equation.
Abstract: In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in a previous paper. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds $\pm \sqrt{2}$, each of which are solutions to a Korteweg-de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimate.
46 citations
TL;DR: The Korteweg-de Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation as discussed by the authors.
Abstract: The fact that the Korteweg-de Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature (see e.g. [17]). In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.
45 citations
TL;DR: In this article, the quadratic B-spline functions and the central difference operator for the time derivative have been used to develop a new algorithm based on the collocation method to solve modified regularized long wave equation.
Abstract: The quadratic B-spline functions and the central difference operator for the time derivative have been used to develop a new algorithm based on the collocation method to solve modified regularized long wave equation. A linear stability analysis of the scheme is shown to be marginally stable. The method is validated by studying solitary wave motion, two and three solitary wave interaction, the evolution of solitary waves, and undular bore development.
44 citations
TL;DR: In this paper, the existence, stability, and instability of periodic travelling wave solutions related to the critical Kortewegde Vries equation u t + 5 u 4 u x + u x x x = 0.
44 citations
TL;DR: Complex Modified Korteweg–deVries Equation is solved numerically using differential quadrature method based on cosine expansion using discrete root mean square error norm L 2 and maximum error normL ∞ for the motion of single solitary wave since it has an analytical solution.
34 citations
TL;DR: In this article, the travelling wave solutions of a special CH-DP equation are studied by using the method of dynamical systems, and exact explicit parametric representations of smooth solitary waves, solitary cusp waves, breaking waves and uncountably infinitely many smooth periodic wave solutions are given.
Abstract: By using the method of dynamical systems, the travelling wave solutions of a special CH–DP equation are studied. Exact explicit parametric representations of smooth solitary waves, solitary cusp waves, breaking waves and uncountably infinitely many smooth periodic wave solutions are given. In different regions of the parametric plane, different phase portraits are determined. The so called loop soliton solution is discussed.
TL;DR: The traveling wave solutions for a generalized coupled KdV equations are discussed and exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.
Abstract: By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.
TL;DR: In this paper, a class of particular travelling wave solutions of the generalized BBM was studied systematically using the factorization technique, and the general travelling wave solution of the BBM and its modified version were also recovered.
Abstract: A class of particular travelling wave solutions of the generalized Benjamin–Bona–Mahony equation is studied systematically using the factorization technique. Then, the general travelling wave solutions of Benjamin–Bona–Mahony equation, and of its modified version, are also recovered.
TL;DR: In this paper, the propagation of weakly nonlinear waves in such a fluid-filled elastic tube, by employing the reductive perturbation method, was studied, and the evolution equation was obtained as variable coefficients modified KdV equation.
Abstract: In the present work, treating the arteries as a thin walled prestressed elastic tube with variable radius, and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube, by employing the reductive perturbation method. By considering the blood as an incompressible non-viscous fluid, the evolution equation is obtained as variable coefficients Korteweg–de Vries equation. Noticing that for a set of initial deformations, the coefficient characterizing the nonlinearity vanish, by re-scaling the stretched coordinates we obtained the variable coefficient modified KdV equation. Progressive wave solution is sought for this evolution equation and it is found that the speed of the wave is variable along the tube axis.
TL;DR: Based on the Navier-Stokes (N-S) equations for viscous, incompressible fluid and the VOF method, 2D and 3D Numerical Wave Tanks (NWT) for nonlinear shallow water waves are built as discussed by the authors.
Abstract: Based on the Navier-Stokes (N-S) equations for viscous, incompressible fluid and the VOF method, 2-D and 3-D Numerical Wave Tanks (NWT) for nonlinear shallow water waves are built. The dynamic mesh technique is applied, which can save computational resources dramatically for the simulation of solitary wave propagating at a constant depth. Higher order approximation for cnoidal wave is employed to generate high quality waves. Shoaling and breaking of solitary waves over different slopes are simulated and analyzed systematically. Wave runup on structures is also investigated. The results agree very well with experimental data or analytical solutions.
TL;DR: In this paper, a velocity potential which is compatible with the Korteweg-de Vries regime and describes a nontrivial fluid flow is proposed, and some qualitative aspects of the flow are discussed.
Journal Article•
TL;DR: In this article, the authors considered a class of semi-linear wave equations with a small parameter and nonlinearities which provide exact kink-antikink type solutions and obtained sucient conditions for the nonlinearity under which the kink collision occurs without changing the waves shape and with only some shifts of the solitary wave trajectories.
Abstract: We consider a class of semi-linear wave equations with a small parameter and nonlinearities which provide the equations having exact kink- type solutions As a main result we to obtain sucient conditions for the nonlinearities under which the kink-antikink collision occurs without changing the waves shape and with only some shifts of the solitary wave trajectories
TL;DR: In this paper, a numerical model was established for simulating water wave dynamic problems by adopting the Smoothed Particle Hydrodynamics (SPH) methods of iterative solution of Poisson's equation for pressure field, and meanwhile the sub-grid turbulence model was applied in the simulation so as to more accurately describe the turbulence characteristics at the time of wave breaking.
Abstract: A numerical model was established for simulating water wave dynamic problems by adopting the Smoothed Particle Hydrodynamics (SPH) methods of iterative solution of Poisson's equation for pressure field, and meanwhile the sub-grid turbulence model was applied in the simulation so as to more accurately describe the turbulence characteristics at the time of wave breaking. In this article, simulation of the problem of the dam collapsing verifies the compoting accuracy of this method, and its results can be identical with the results of VOF method and the experimental results by comparison. Numerical simulations of the course of solitary wave and cnoidal wave run-up breaking on beaches were conducted, and the results are basically consistent with experimental results. This indicates that the SPH method is effective for the numerical simulation of the complex problems of water wave dynamics.
TL;DR: In this article, the binary F -expansion method with two wave speeds has been introduced based on the traditional F-expansion, and using this new extend method and a simple transformation technique, the n ǫ+ǫ-1)-dimensional sine-Gordon equation was studied.
01 Jan 2009
TL;DR: In this paper, the authors deal with linear wave equations on Lorentzian manifolds and recall the physical origin of that equation which describes the propagation of a wave in space.
Abstract: This chapter deals with linear wave equations on Lorentzian manifolds. We first recall the physical origin of that equation which describes the propagation of a wave in space.
TL;DR: In this paper, an attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg-de Vries (KdV) equation by using tanh-method or hyperbolic method.
Abstract: An attempt has been made to obtain exact analytical traveling wave solution or simple wave solution of higher-order Korteweg–de Vries (KdV) equation by using tanh-method or hyperbolic method. The higher-order equation can be derived for magnetized plasmas by using the reductive perturbation technique. It is found that the exact solitary wave solution of higher-order KdV equation is obtained by tanh-method. Using this method, different kinds of nonlinear wave equations can be evaluated. The higher-order nonlinearity and higher-order dispersive effect can be observed from the solutions of the equations. The method is applicable for other nonlinear wave equations.
TL;DR: The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory by derived within different parameter regions of the parameter space.
Abstract: The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.
TL;DR: In this paper, the problem of wave propagation in microstructured materials, characterized by higher-order nonlinear and higherorder dispersive effects, is solved analytically by exact methods.
Abstract: The problems under consideration are related to wave propagation in microstructured materials, characterized by higher-order nonlinear and higher-order dispersive effects; particularly, the wave propagation in dilatant granular materials. In the present paper the model equation is solved analytically by exact methods. The types of solutions are defined and discussed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). The dispersion properties and the relation between group and phase velocities of the model equation are studied. The diagrams are drawn to illustrate the physical properties of the exact solutions.
TL;DR: In this paper, a new coupling model of wave interaction with porous medium is established in which the wave field solver is based on the two dimensional Reynolds Averaged Navier-Stokes (RANS) equations with a ke closure.
TL;DR: The periodic wave solutions for the system of coupled Korteweg-de Vries equations are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained.
TL;DR: In this article, the authors have studied obliquely propagating dust-acoustic nonlinear periodic waves in a magnetized dusty plasma consisting of electrons, ions, and dust grains with variable dust charge.
Abstract: We have studied obliquely propagating dust-acoustic nonlinear periodic waves, namely, dust-acoustic cnoidal waves, in a magnetized dusty plasma consisting of electrons, ions, and dust grains with variable dust charge. Using reductive perturbation method and appropriate boundary conditions for nonlinear periodic waves, we have derived Korteweg–de Vries (KdV) equation for the plasma. It is found that the contribution to the dispersion due to the deviation from plasma approximation is dominant for small angles of obliqueness, while for large angles of obliqueness, the dispersion due to magnetic force becomes important. The cnoidal wave solution of the KdV equation is obtained. It is found that the frequency of the cnoidal wave depends on its amplitude. The effects of the magnetic field, the angle of obliqueness, the density of electrons, the dust-charge variation and the ion-temperature on the characteristics of the dust-acoustic cnoidal wave are also discussed. It is found that in the limiting case the cnoi...
01 Jan 2009
TL;DR: In this paper, the authors extended the Jacobi elliptic function expansion method by constructing four new Jacobian elliptic functions, and applied this method to Zakharov equations for illustration, abundant new doubly periodic solutions are obtained, these solutions are degenerated to the solitary wave solutions and the triangle function solutions in the limit cases when the modulus of the Jacobian Elliptic functions m!1 or 0.
Abstract: We extended the Jacobi elliptic function expansion method by constructing four new Jacobian elliptic functions, and apply this method to Zakharov equations for illustration, abundant new doubly periodic solutions are obtained, these solutions are degenerated to the solitary wave solutions and the triangle function solutions in the limit cases when the modulus of the Jacobian elliptic functions m!1 or 0, which shows that the new method is more powerful to seek the exact solutions of the nonlinear partial differential equations in mathematical physics.
TL;DR: In this article, an improved algorithm is devised to derive exact travelling wave solutions of nonlinear differential-difference equations (DDEs) by means of Jacobi elliptic functions.
Abstract: In this paper, an improved algorithm is devised to derive exact travelling wave solutions of nonlinear differential-difference equations (DDEs) by means of Jacobi elliptic functions. With the aid of symbolic computation, we choose the integrable discrete nonlinear Schrodinger equation to illustrate the validity and advantages of the method. As a result, new and more general Jacobi elliptic function solutions are obtained, from which hyperbolic function solutions and trigonometric function solutions are derived when the modulus m→1 and 0. It is shown that the proposed method provides a more effective mathematical tool for nonlinear DDEs in mathematical physics.
TL;DR: In this article, the authors present an analytical study of a high-order acoustic wave equation in one dimension, and reformulate a previously given equation in terms of an expansion of the acoustic Mach number.