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 [, ].

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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

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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 [–].

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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

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[13] T. S. El-Danaf, M. A. Ramadan and F. E. I. AbdAlaal, “The Use of Adomian Decomposition Method for Solv- ing the **Regularized** **Long**-**Wave** **Equation**,” 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 **regularized** **long**-**wave** **equation** 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 [4] and the multiple scale expansion method of Kraenkel et al [6].

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In the current article, the solitary **wave** solutions of the two dimensional **regularized** **long**-**wave** **equation** 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-

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where p is a positive integer. Zhang[23] solved the GRLW **equation** by ﬁnite diﬀerence method for a Cauchy problem. Kaya et.al [24] also studied the GRLW **equation** with Adomian decomposition method. Ramos[25] used quasilinearization method based on ﬁ- nite diﬀerences for solving the GRLW **equation**. Roshan[26] 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 **regularized** **long** **wave** (MRLW) **equation** which is a special form of the GRLW **equation**. Gardner et al.[27] 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[30]. K. R. Raslan and S. M. Hassan[31] 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[32] 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.[33] have designed a numerical scheme based on quartic B-spline collocation method for the numerical solution of MRLW **equation**.

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Abstract. In this paper, we employ a sub-**equation** method to nd the exact solutions to the fractional (1 + 1) and (2 + 1) **regularized** **long**-**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, **Regularized** **long** **wave** 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.

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In this work, our main purpose is to develop of a suﬃciently robust, accurate and eﬃcient numerical scheme for the solution of the **regularized** **long** **wave** (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.

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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 **long** **wave**). 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.

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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 ′

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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.

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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** .

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We would like in this paper to write Dirac **equation** as a second - order differential **equation** in space and time mimicking the standard **wave** **equation**. 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.

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In this paper, the local fractional variational iteration method is given to handle the damped **wave** **equation** and dissipative **wave** **equation** 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.

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