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].

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159-207]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for Bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic arguments, ...

We propose a hierarchy of nonlinearly dispersive generalized Korteweg–de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. It is shown that two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the propo...

In this paper, He’s variational iteration method (VIM) has been used to obtain solutions of the seventh-order Sawada-Kotera equation (sSK) and a Lax’s seventh order KdV equations(LsKdV).The numerical solutions are compared with the Adomian decomposition method(ADM) and the known analytical solutions.The work confirms the power of the VIM in reducing the size of calculations w.r.t. ADM. Some ill...

The Thomas-Fermi (TF) equation has proved to beuseful for the treatment of many physical phenomena. In this pa-per, the traveling wave solutions of the KdV equation is investi-gated by the simplest equation method. Also, the effect of differentparameters on these solitary waves is considered. The numericalresults is conformed the good accuracy of presented method.    

We consider the (KdV)/(KP-I) asymptotic regime for the Nonlinear Schrödinger Equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg-de Vries equation (in dimension 1) and to the Kadomtsev-Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/...

The initial boundary-value problem for the Korteweg–de Vries (KdV) equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter-plane problem, in contrast to the one bo...

The Korteweg-de Vries (KdV) equation is known as a model of long waves in an infinitely long canal over a flat bottom and approximates the two-dimensional water wave problem, which is a free boundary problem for incompressible Euler equation with the irrotational condition. In this paper, we consider the validity of this approximation in the case of presence of surface tension. Moreover, we con...