In this paper we use the modified tanh-coth method to solve the Kawahara and the modified Kawahara equations. New multiple traveling wave solutions are obtained for the Kawahara and the modified Kawahara equations. 2000 Mathematics Subject Classification: 35K01; 35J05.

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravitycapillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in H(R) with s > − 4 and the local well-posedness for the modified Kawahara equation in H(...

The Kawahara and modified Kawahara equations are fifth-order KdV type equations and have been derived to model many physical phenomena such as gravitycapillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness of the initial-value problem for Kawahara equation in H(R) with s > − 4 and the local well-posedness for the modified Kawahara equation in H(...

In this paper, we apply the Exp-function method to Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara equation is the combination of the Rosenau and standard Kawahara equations and Rosenau-KdV equation is the combination of the Rosenau and standard KdV equations. These equations are nonlinear partial differential equations (NPDE) which play an important role in mathematical physics. E...

An efficient and accurate numerical scheme is proposed, analyzed and implemented for the Kawahara and modified Kawahara equations which model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. The scheme consists of dual-Petrov-Galerkin method in space and Crank-Nicholson-leap-frog in time such that at each time step only a sparse banded linear sys...

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com. Appl. Math. 233 (2009) 97–102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong.

This paper is concerned with the Cauchy problem of the modified Kawahara equation. By using the Fourier restriction norm method introduced by Bourgain, and using the I-method as well as the L 2 conservation law, we prove that the modified Kawahara equation is globally well-posed for the initial data in the Sobolev space H s (R) with s > − 3 22 .

Exact solutions of the Kawahara equation by Assas [L.M.B. Assas, J. Com. Appl. Math. 233 (2009) 97–102] are analyzed. It is shown that all solutions do not satisfy the Kawahara equation and consequently all nontrivial solutions by Assas are wrong. Recently Assas in [1] looked for exact solutions of the Kawahara equation using the Exp function method. This equation can be written as ut + α uux +...

In this paper, we study a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain T (a unit circle in the plane) with an internal control acting on an arbitrary small nonempty subdomain of T. Aided by the Bourgain smoothing property of the Kawahara equation on a periodic domain, we show that the system is locally exactly controllable and expon...

In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation ut + uux + uxxx − γ1uxxxxx = 0, and the modified Kawahara-KdV equation ut + 3u 2ux + uxxx − γ2uxxxxx = 0, where γi ∈ R is a positive number when i = 1, 2. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert in [1].