lution at diﬀerent times for both cases are drawn in Figure and Figure for the RLW equation, and Figure and Figure for the MRLW equation, respectively. Furthermore, the quantities for three conservative laws are presented in Table . As seen from these results, an initial date with Maxwellian disturbance will resolve into a sequence of solitary waves in the stable range ordered by amplitude with the larger waves in the front, followed by a dispersive tail. Clearly, the presented results are consistent with earlier work on this topic in [, ].
where Q() and E() are two positive constants which relate to the initial condition. Existence and uniqueness of the solution of the RLW equation are given in . Its analytical solution was found  under restricted initial and boundary conditions, and, therefore, it became interesting from a numerical point of view. Some numeri- cal methods for the solution of the RLW equation such as variational iteration method [, ], ﬁnite-diﬀerence method [–], Fourier pseudospectral method , ﬁnite element method [–], collocation method  and adomian decomposition method  have been introduced in many works. In , Li and Vu-Quoc pointed out that ‘in some ar- eas, the ability to preserve some invariant properties of the original diﬀerential equa- tion is a criterion to judge the success of a numerical simulation.’ Meanwhile, Zhang et al.  thought that the conservative diﬀerence schemes perform better than the non- conservative ones, and the non-conservative diﬀerence schemes may easily show non- linear ‘blow-up.’ Hence, constructing a conservative diﬀerence scheme for the numerical solution of the nonlinear partial diﬀerential equation is quite signiﬁcant. In this paper, coupled with the Richardson extrapolation, a two-level nonlinear Crank-Nicolson ﬁnite diﬀerence scheme for problems (.)-(.), which has the accuracy of O(τ + h ) without
In , Zhang considered a linear conservative scheme for GRLW equation, however, the accuracy of the scheme is only second-order. Recently, there has been growing in- terest in high-order compact methods to solve the partial diﬀerential equations [– ], where fourth-order compact ﬁnite diﬀerence approximation solutions for the tran- sient wave equations, a N-carrier system, the Klein-Gordon equation, the Sine-Gordon equation, the one-dimensional heat and advection-diﬀusion equations, the Schrödinger equation, the Klein-Gordon-Schrödinger equation and the RLW equation were shown, respectively. These numerical methods may give us many enlightenments to design a new numerical scheme for the GRLW equation. For a wide and most complete vision concern- ing the importance, the breadth, and the interest of the topics covered, we should also recall the study done on the long waves in [–].
of the invariants obtained by the present method with those obtained in Refs. [, , , ] are listed in Table . It is seen that the obtained values of the invariants remain almost constant during the computer run. The development of the interaction of two soli- tary waves is shown in Figure . It can be seen from the ﬁgure that at t = the wave with larger amplitude is to the left of the second wave with smaller amplitude. Since the taller wave moves faster than the shorter one, it catches up and collides with the shorter one at t = and then moves away from the shorter one as time increases. At t = , the am- plitude of larger waves is . at the point x = . whereas the amplitude of the smaller one is . at the point x = . It is found that the absolute diﬀerence in am- plitude is . × – for the smaller wave and . × – for the larger wave for this
 T. S. El-Danaf, M. A. Ramadan and F. E. I. AbdAlaal, “The Use of Adomian Decomposition Method for Solv- ing the RegularizedLong-WaveEquation,” Chaos, Soli- tons & Fractals, Vol. 26, No. 3, 2005, pp. 747-757. doi:10.1016/j.chaos.2005.02.012
the power and elegance of the present method, we compared our result with the exact travelling wave solution of the symmetric regularizedlong-waveequation with quadratic nonlinearity. These results show that for weakly nonlinear case the solutions for both approaches coincide with each other. The present method is seen to be fairly simple as compared to the renormalization method of Kodama and Taniuti  and the multiple scale expansion method of Kraenkel et al .
Equation 1.3 is explicitly symmetric in the x and t derivatives and is very similar to the regularizedlongwaveequation that describes shallow water waves and plasma drift waves 2, 3 . The SRLW equation also arises in many other areas of mathematical physics 4–6 . Numerical investigation indicates that interactions of solitary waves are inelastic 7 ; thus, the solitary wave of the SRLWE is not a solution. Research on the wellposedness for its solution and numerical methods has aroused more and more interest. In 8 , Guo studied the existence, uniqueness, and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In 9 , Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite diﬀerence method for the initial-boundary value problem of SRLWEs see 9– 15 .
In the current article, the solitary wave solutions of the two dimensional regularizedlong-waveequation in plasma and rotating flows simulated by using extended mapping method, and we hope these solitary waves are helpful to understand the nonlinear phenomena described by the resonant Davey-Stewartson equation in the fields like capillarity fluids. We have presented the extended mapping method to construct more general exact solu-
where p is a positive integer. Zhang solved the GRLW equation by ﬁnite diﬀerence method for a Cauchy problem. Kaya et.al  also studied the GRLW equation with Adomian decomposition method. Ramos used quasilinearization method based on ﬁ- nite diﬀerences for solving the GRLW equation. Roshan solved the GRLW equation numerically by the Petrov-Galerkin method using a linear hat function as the trial func- tion and a quintic B-spline function as the test function. In this paper, we consider the modiﬁed regularizedlongwave (MRLW) equation which is a special form of the GRLW equation. Gardner et al. have developed a collocation solution to the MRLW equation using quintic B-splines ﬁnite elements. A. K. Khalifa et al.[28, 29] obtained the numer- ical solutions of the MRLW equation using ﬁnite diﬀerence method and cubic B-spline collocation ﬁnite element method. Solutions based on collocation method using quadratic B-spline ﬁnite elements and the central ﬁnite diﬀerence method for time are investigated by K. R. Raslan. K. R. Raslan and S. M. Hassan have solved the MRLW equa- tion by a collocation ﬁnite element method using quadratic, cubic, quartic and quintic B- splines to obtain the numerical solutions of the single solitary wave. S. B. Gazi Karakoc and T.Geyikli has solved the equation by Petrov-Galerkin method in which the ele- ment shape functions are cubic and weight functions are quadratic B-splines. Fazal-i-Haq et al. have designed a numerical scheme based on quartic B-spline collocation method for the numerical solution of MRLW equation.
Abstract. In this paper, we employ a sub-equation method to nd the exact solutions to the fractional (1 + 1) and (2 + 1) regularizedlong-wave equations which arise in several physical applications, including ion sound waves in plasma, by using a new denition of fractional derivative called conformable fractional derivative. The presented method is more eective, powerful, and straightforward and can be used for many other nonlinear partial fractional dierential equations.
It is known the conservative scheme is better than the nonconservative ones. Zhang et al. 1 point out that the nonconservative scheme may easily show nonlinear blow up. In 2 Li and Vu-Quoc said “. . . in some areas, the ability to preserve some invariant properties of the original diﬀerential equation is a criterion to judge the success of a numerical simulation”. In 3–11 , some conservative finite diﬀerence schemes were used for a system of the generalized nonlinear Schr ¨odinger equations, Regularizedlongwave RLW equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative diﬀerence scheme for the general Rosenau-RLW equation, which simulates conservative laws 1.4 and 1.5 at the same time. The outline of the paper is as follows. In Section 2, a nonlinear diﬀerence scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
wave amplitude and a large wave length in nonlinear dispersive and many other physical systems. Then, the idea of Equal Width (EW) waveequation, which has both positive and negative amplitudes with the same width, was proposed by Morrison et al. . Therefore, the Generalized RegularizedLongWave (GRLW) equation and the Generalized Equal Width (GEW) waveequation oer some technical advantages over the Generalized Korteweg-de Vries (GKdV) equa- tion. Such types of wave equations have solitary wave solutions, which are pulse-like.
In this work, our main purpose is to develop of a suﬃciently robust, accurate and eﬃcient numerical scheme for the solution of the regularizedlongwave (RLW) equation, an important partial diﬀerential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semi-discretization and the backward diﬀerence formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments conﬁrm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently.
Equation (7) has well been known as DLWE . In the concept of the spectral transform, Eq. (7) is analyzed in  and thereafter in , where it was related to the Schrödinger equation with linear spectral dependence in the potential. In hydrodynamics it exhibits the evolution of the horizontal velocity component of water waves propagating in both directions in an inﬁnite narrow channel of constant depth and could be generated from the water wave equations by including one more order of nonlinearity than is done in deriving the Boussinesq equation. The integrability and derivation for Eq. (7) was presented in . Furthermore, in  the authors related it directly to the spectral problem
In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differen- tial equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional dispersive longwave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlin- ear ordinary differential equations which is solved via the shooting method, coupled with Runge-kutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.
The backward wave is the function Q (x + t) which represents a wave travelling in the negative x-direction with scaled speed 1. The wave shape is determined by Q (x) and the value of the wave is constant along the lines x + t = const (in physical variables, x ′
The growing interest of nonlinear waves has been given to the propagation in the dynamical system. The solitary wave ansatz method is rather heuristic and processes significant features that make it practical for the determina- tion of single soliton solutions for a wide class of nonlinear evolution equations. The solitary wave and shock wave solitons have been constructed, using the solitary wave ansatz method, for Ostrovsky equation and Poten- tial Kadomstev-Petviashvili equation and we clearly see the consistency, which has recently been applied suc- cessfully.
A problem is said to be well-posed for a given equation, if for any assigned data, there exists exactly one corresponding solution. It is simple to observe that problem (i) is well-set for the Laplace equation in n(n> 3) variables while this does not hold for the general linear elliptic equation .
We would like in this paper to write Dirac equation as a second - order differential equation in space and time mimicking the standard waveequation. In doing so, we have found new space and time transformations under which Dirac equation represents a particle with zero mass. Moreover, the ordinary Dirac equation is invariant under these transformations. Interestingly, the invariance of Klein-Gordon equation under these transformation yields Dirac equation. The new Dirac equation has many interesting consequences. As in the de Broglie theory, the electron is described by a wavepacket whose group velocity is equal to the speed of light in vacuum. The nature of this wave may explain the Zitterbewegung ex- hibited by the electron as found by Schrodiner.
In this paper, the local fractional variational iteration method is given to handle the damped waveequation and dissipative waveequation in fractal strings. The approximation solutions show that the methodology of local fractional variational iteration method is an eﬃcient and simple tool for solving mathematical problems arising in fractal wave motions.