The family of the KdV equations, the most famous equations embodying both nonlinearity and dispersion, has attracted enormous attention over the years and has served as the model equation for the development of soliton theory. In this paper we present a comparative study between two different methods for solving the general KdV equation, namely the numerical Crank Nicolson method, and the semi-...

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 derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether’s ...

we show that a recently introduced lax pair of the sawada-kotera equation is nota new one but is trivially related to the known old lax pair. using the so-called trivialcompositions of the old lax pairs with a differentially constrained arbitrary operators,we give some examples of trivial lax pairs of kdv and sawada-kotera equations.

Elzaki transform and Adomian polynomial is used to obtain the exact solutions of nonlinear fifth order Korteweg-de Vries (KdV) equations. In order to investigate the effectiveness of the method, three fifth order KdV equations were considered. Adomian polynomial is introduced as an essential tool to linearize all the nonlinear terms in any given equation because Elzaki transform cannot handle n...

We derive a Riemann–Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the nume...

It is known that some periodic solutions of the complex KdV equation with smooth initial data blow up in finite time. In this paper, we investigate the effect of dissipation on the regularity of solutions of the complex KdV equation. It is shown here that if the initial datum is comparable to the dissipation coefficient in the L2-norm, then the corresponding solution does not develop any finite...

The Virasoro-Bott group endowed with the right-invariant L2metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.