Solutions of a boundary value problem for the Korteweg–de Vries equation are approximated numerically using a finite-difference method, and a collocation method based on Chebyshev polynomials. The performance of the two methods is compared using exact solutions that are exponentially small at the boundaries. The Chebyshev method is found to be more efficient. © 2009 IMACS. Published by Elsevier...

Abstract In this study, we present two different methods a sech-tanh method and extended tanh-method to obtained the soliton solutions of the two-dimensional Korteweg-de Vries-Burgers (KdVB) equation with the initial conditions. These solutions include bright and dark solitary wave solutions, triangular solutions and complex line soliton wave solution. These solutions are stable and have applic...

Abstract We prove local existence and uniqueness of solutions of the focusing modified Korteweg de Vries equation ut+u 2 ux+uxxx = 0 in classes of unbounded functions that admit an asymptotic expansion at infinity in decreasing powers of x. We show that an asymptotic solution differs from a genuine solution by a smooth function that is of Schwartz class with respect to x and that solves a gener...

Linear and non-linear surface waves on a ferrofluid cylinder surrounding a current-carrying wire are investigated. Suppressing the Rayleigh-Plateau instability of the fluid column by the magnetic field of a sufficiently large current in the wire axis-symmetric surface deformations are shown to propagate without dispersion in the long wavelength limit. Using multiple scale perturbation theory th...

A method is given for finding the shifts in position of the solitons for the case of nonzero reflection coefficient. Expressions for boost generators in terms of scattering data play a prominent role in the analysis. Phase-shift formulas which show the effect of the radiation component on the soliton motion are deduced for the nonlinear Schrodinger equation, the Korteweg —de Vries equation, and...

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...

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)

In this paper, an efficient numerical schemes based on the Haar wavelet method are applied for finding numerical solution of nonlinear third-order modified Korteweg-de Vries (mKdV) equation as well as modified Burgers’ equations. The numerical results are then compared with the exact solutions. The accuracy of the obtained solutions is quite high even if the number of calculation points is small.