In this paper we investigate a nonlinear evolution model described by the Rosenau-KdV equation. We propose a three-level average implicit finite difference scheme for its numerical solutions and prove that this scheme is stable and convergent in the order of O(τ2 + h2). Furthermore we show the existence and uniqueness of numerical solutions. Comparing the numerical results with other methods in...

This paper carries out the integration of Burgers equation by the aid of tanh method. This leads to the complex solutions for the Burgers equation, KdV–Burgers equation, coupled Burgers equation and the generalized time-delayed Burgers equation. Finally, the perturbed Burgers equation in (1+1) dimensions is integrated by the ansatz method. 2010 Elsevier Inc. All rights reserved.

The matrix 2x2 spectral differential equation of the second order is considered on x in (−∞, +∞). We establish elementary Darboux transformations covariance of the problem and analyze its combinations. We select a second covariant equation to form Lax pair of a coupled KdV-MKdV system. The sequence of the elementary Darboux transformations of the zero-potential seed produce two-parameter soluti...

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 1, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order ǫ. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. In...

Recent advances in the numerical solution of Riemann–Hilbert problems allow for the implementation of a Cauchy initial value problem solver for the Korteweg–de Vries equation (KdV) and the defocusing modified Korteweg–de Vries equation (mKdV), without any boundary approximation. Borrowing ideas from the method of nonlinear steepest descent, this method is demonstrated to be asymptotically accur...

We study the stability of spatially periodic solutions to the Kawahara equation, a fifth order, nonlinear partial differential equation. The equation models the propagation of nonlinear water-waves in the long-wavelength regime, for Weber numbers close to 1/3 where the approximate description through the Korteweg-de Vries (KdV) equation breaks down. Beyond threshold, Weber number larger than 1/...

We construct N = 4 supersymmetric KdV equation as a hamiltonian flow on the N = 4 SU(2) super Virasoro algebra. The N = 4 KdV superfield, the hamiltonian and the related Poisson structure are concisely formulated in 1D N = 4 harmonic superspace. The most general hamiltonian is shown to necessarily involve SU(2) breaking parameters which are combined in a traceless rank 2 SU(2) tensor. First non...

The new generalized Harry Dym equation of Z. Popowicz is transformed into the Hirota–Satsuma system of coupled KdV equations.

We study solitary wave interactions in the Euler–Poisson equations modeling ion acoustic plasmas and their approximation by KdV n-solitons. Numerical experiments are performed and solutions compared to appropriately scaled KdV n-solitons. While largely correct qualitatively the soliton solutions did not accurately capture the scattering shifts experienced by the solitary waves. We propose corre...

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation ut + 6uux + ǫ uxxx = 0, u(x, t = 0, ǫ) = u0(x), for ǫ small, near the point of gradient catastrophe (xc, tc) for the solution of the dispersionless equation ut + 6uux = 0. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a hi...