This paper investigates the implementation of Variational Iteration Method (VIM) to practical and higher order nonlinear equations in kind of Korteweg-de-Vries (KdV) equation. The obtained solutions from thirdand fourth-order modified KdV are compared with the exact and Homotopy Perturbation Method (HPM) solutions. Results illustrate the efficiency and capability of VIM to solve high order nonl...

We extend Halphen’s theorem which characterizes the solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n×n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with comple...

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type with time-delayed internal feedback. Under suitable assumptions on time delay coefficients, we prove that solutions of this system are exponentially stable. First, considering damping and delayed system, some restriction spatial length domain, energy goes to $ 0 as t \rightarrow \infty $. After that, by introducing more...

Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solv...

We derive the modulation equations or Whitham equations for the Camassa– Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir...

In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develo...

In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develo...