For a < 0 one obtains solitary patterns having cusps or infinite slopes [2]. They discovered solitary waves, called compactons, with a compact support characterized by the absence of infinite wings or the absence of infinite tails. If a = 1, then (1) has a focusing (+) branch that exhibits compacton solutions. If a = −1, then (1) has a defocusing (−) branch that exhibits solitary pattern soluti...

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(...

Scaling similarity solutions of three integrable PDEs, namely the Sawada-Kotera, fifth order KdV and Kaup-Kupershmidt equations, are considered. It is shown that the resulting ODEs may be written as non-autonomous Hamiltonian equations, which are time-dependent generalizations of the well-known integrable Hénon-Heiles systems. The (time-dependent) Hamiltonians are given by logarithmic derivativ...

In this work we apply an extended hyperbolic function method to solve the nonlinear family of third order Korteweg de-Vries (KdV) equations, namely, the KdV equation, the modified KdV (mKdV) equation, the potential KdV (pKdV) equation, the generalized KdV (gKdV) equation and gKdV with two power nonlinearities equation. New exact travelling wave solutions are obtained for the KdV, mKdV and pKdV ...

In a previous paper we proved that long-wavelength solutions of the waterwave problem in the case of zero surface tension split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries (KdV) equation. In this paper we examine the effect of surface tension on this scenario. We find that we o...

A study is presented of fully discretized lattice equations associated with the KdV hierarchy. Loop group methods give a systematic way of constructing discretizations of the equations in the hierarchy. The lattice KdV system of Nijhoff et al. arises from the lowest order discretization of the trivial, lowest order equation in the hierarchy, bt = bx. Two new discretizations are also given, the ...

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...