Exp-function method for traveling wave solutions of nonlinear evolution equations

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations. The proposed technique is tested on the modified Zakharov-Kuznetsov (ZK) and Zakharov-Kuznetsov-Modified-Equal-Width (ZK-MEW) equations. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed exp-function method.     

Weak solutions to quasilinear wave equations of Klein-Gordon or Sine-Gordon type and relaxation to reaction-diffusion equations

This paper regards the existence of weak solutions for a quasilinear wave equation of Klein-Gordon and Sine-Gordon type with the presence of a linear damping term and the relaxation to the reaction-diffusion equation when the momentum relaxation time tends to zero. In the limit process is fundamental the celebrated Div-curl Lemma of Tartar and Murat. Received February 5, 1996     

Application of the Exp-function method for nonlinear differential-difference equations

In this paper, we established abundant travelling wave solutions for some nonlinear differential-difference equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics.     

New travelling wave solutions to the Ostrovsky equation

In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods.     

New periodic solutions for nonlinear evolution equations using Exp-function method

The Exp-function method is used to obtain generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics using symbolic computation. The method is straightforward and concise, and its applications are promising.     

New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method

Using symbolic computation, we apply the Exp-function method to construct new kinds of solutions to the foam drainage equation. We demonstrate that the Exp-function method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

Symmetry analysis and exact solutions of some Ostrovsky equations

We apply the classical Lie method and the nonclassical method to a generalized Ostrovsky equation (GOE) and to the integrable Vakhnenko equation (VE), which Vakhnenko and Parkes proved to be equivalent to the reduced Ostrovsky equation. Using a simple nonlinear ordinary differential equation, we find that for some polynomials of velocity, the GOE has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves, solitary waves, compactons, etc. The nonclassical method applied to the associated potential system for the VE yields solutions that arise from neither nonclassical symmetries of the VE nor potential symmetries. Some of these equations have interesting behavior such as “nonlinear superposition.”     

New solitonary solutions for modified forms of DP and CH equations using Exp-function method

In this paper we use the Exp-function method for the analytic treatment of the modified forms of Degasperis-Procesi and Camassa-Holm equations. New solitonary solutions are formally derived. The change of the parameters, that in its turn drastically changes the characteristics of the equations, is examined. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving high-dimensional nonlinear evolutions in mathematical physics.     

New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations

In this letter, the Kaup–Kupershmidt, (2+1)$(2+1)$-dimensional Potential Kadomtsev–Petviashvili (shortly PKP) equations are presented and the Exp-function method is employed to compute an approximation to the solution of nonlinear differential equations governing the problem. It has been attempted to show the capabilities and wide-range applications of the Exp-function method. This method can be used as an alternative to obtain analytic and approximate solution of different types of differential equations applied in engineering mathematics.     

Positon and positon-like solutions of the Korteweg-de Vries and Sine-Gordon equations

It is shown that the positon solution, reported recently for the Korteweg-de Vries and other completely integrable equations, can be regarded as a limiting case of the well-known 2-soliton formula. The existence of integrals of motion related to singular positon solutions is also discussed.     

Application of Exp-function method to generalized Burgers-Fisher equation

In this paper,the Exp-function method is used to construct exact solitary wave solutions for the generalized Burgers-Fisher equation with nonlinear terms of any order.With the aid of Maple computation,we obtain many new and more general exact solitary wave solutions expressed by various exponential and hyperbolic functions.Our results can successfully recover previously known solitary wave solutions that have been found by the tanh-function method and other more sophisticated methods.     

Exp-function method for the exact solutions of Sawada-Kotera equation

In this paper, we use the Exp-Function method to construct some exact solutions of generalized Sawada-Kotera equation. It is shown that the Exp-Function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

Newton's method and periodic solutions of nonlinear wave equations

We prove the existence of periodic solutions of the nonlinear wave equation satisfying either Dirichlet or periodic boundary conditions on the interval [O, π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.     

A note on “New travelling wave solutions to the Ostrovsky equation”

In a recent paper by Ya?ar [E. Ya?ar, New travelling wave solutions to the Ostrovsky equation, Appl. Math. Comput. 216 (2010), 3191-3194], ‘new’ travelling-wave solutions to the transformed reduced Ostrovsky equation are presented. In this note it is shown that some of these solutions are disguised versions of known solutions.     

Exact compacton and generalized kink wave solutions of the extended reduced Ostrovsky equation

The extended reduced Ostrovsky equation (EX-ROE) are investigated by using the bifurcation method of planar systems and simulation method of differential equations. The bifurcation phase portraits are drawn in different regions of parameter plane. The planar graphs of the compactons and the generalized kink waves are simulated by using software Maple. Exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink wave solutions are given. The dynamic behavior of these solutions are also investigated.     

Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: utt=A(x)(Dx(u)ux)+B(x)Q(u), Ax≠0. A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.     

New exact solutions for the Kawahara equation using Exp-function method

In this paper, we applied the Exp-function method to solve the Kawahara equation. This method can be used to obtain new exact solutions and periodic solutions with parameters are obtained. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful (mathematical tools) for discrete nonlinear evolution equations in mathematical physics.     

Asymptotics of solutions to the generalized Ostrovsky equation

We consider the Cauchy problem for the generalized Ostrovsky equation

_utx=u+_(f(u))xx,$u_{tx}=u+{(f(u))}_{xx},$

where f(u)=|u|^ρ−1u$f(u)={|u|}^{\rho -1}u$ if ρ   is not an integer and f(u)=u^ρ$f(u)=u^{\rho }$ if ρ   is an integer. We obtain the L^∞${\mathbf{L}}^{\infty }$ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity.     

Exact solutions of the Drinfel’d-Sokolov-Wilson equation using the Exp-function method

The generalized solitary solutions of the classical Drinfel’d-Sokolov-Wilson equation (DSWE) are obtained using the Exp-function method. Then, some of these solutions are easily converted into kink-shaped solutions and blow-up solutions by a simple transformation.     

Application of Exp-function method to potential Kadomtsev–Petviashvili equation

Exact periodic kink-wave solution, periodic soliton and doubly periodic solutions for the potential Kadomtsev–Petviashvii (PKP) equation are obtained using Exp-function method with the help of Maple computation.