The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure i...

In Section 1 we present a general principle for associating nonlinear equations of evolutions with linear operators so that the eigenvalues of the linear operator are integrals of the nonlinear equation. A striking instance of such a procedure is the discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation. In Secti...

We prove global existence and modified scattering for the solutions of Cauchy problem to fractional Korteweg-de Vries equation with cubic nonlinearity small, smooth localized initial data.

Stationary wave solutions of the perturbed Korteweg-de Vries equation are considered in the presence of external hamiltonian perturbations. Conditions of their chaotic behaviour are studied with the help of Melnikov theory. For the homoclinic chaos Poincaré sections are constructed to demonstrate the complicated behaviour, and Lyapunov exponents are also numerically calculated.

extended Korteweg-de Vries equation Bernard K. Ee, a) R. H. J. Grimshaw, b) K. W. Chow, c) and D-H. Zhang d) Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hon...

Convergence and quasi-optimality of an adaptive finite element method for controlling L2 errors ⋆ Alan Demlow1, Rob Stevenson2 1 Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY 40506–0027 ([email protected]) 2 Korteweg-de Vries (KdV) Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands (R.P.Stevenson@...

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In par...

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval ut + ux + uxxx + uux = 0, u(x, 0) = φ(x), 0 < x < L, t > 0 (0.1) subject to the nonhomogeneous boundary conditions, B1u = h1(t), B2u = h2(t), B3u = h3(t) t > 0 (0.2)