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...

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...

We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negati...

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equa...

This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt ...

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces evolution apply traveling wave setting to study one of these Assuming a particular type solution leads us solve cubic equation. New solutions are found, but none classical solitary solutions.

Abstract: In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applie...

  In this paper, we establish exact solutions for the time-fractional Klein-Gordon equation, and the time-fractional Hirota-Satsuma coupled KdV system. The He’s semi-inverse and the Kudryashov methods are used to construct exact solutions of these equations. We apply He’s semi-inverse method to establish a variational theory for the time-fractional Klein-Gordon equation, and the time-fractiona...