Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations 15 (2002) 1325–1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of...

It is known that a transform of Liouville type allows one to pass from an equation of the Korteweg-de Vries (K-dV) hierarchy to a corresponding equation of the Camassa-Holm (CH) hierarchy [2, 39]. We give a systematic development of the correspondence between these hierarchies by using the coefficients of asymptotic expansions of certain Green’s functions. We illustrate our procedure with some ...

In this paper we present a set of results on the integration and on the symmetries of the lattice potential Korteweg-de Vries (lpKdV) equation. Using its associated spectral problem we construct the soliton solutions and the Lax technique enables us to provide infinite sequences of generalized symmetries. Finally, using a discrete symmetry of the lpKdV equation, we construct a large class of no...

We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that using the classical spectral element method, i.e. a high order continuous Galerkin approximation, it is possible to develop satisfactory schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invar...

We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations. We then discuss when the associated linearized equation admits solutions exponentiall...

Based on quantum hydrodynamics theory (QHD), the propagation of nonlinear quantum dust-ion acoustic (QDIA) solitary waves in a ‎collision-less, unmagnetized four component quantum plasma consisting of electrons, positrons, ions and stationary negatively charged ‎dust grains with dust charge variation is investigated using reductive perturbation method. The charging current to the dust grains ca...

The existence of "dispersion-managed solitons," i.e., stable pulsating solitary-wave solutions to the nonlinear Schrodinger equation with periodically modulated and sign-variable dispersion is now well known in nonlinear optics. Our purpose here is to investigate whether similar structures exist for other well-known nonlinear wave models. Hence, here we consider as a basic model the variable-co...

R esum e. La propagation unidirectionnelle d'ondes de faible amplitude et de grande longueur d'onde est d ecrite, dans de nombreux syst emes physiques, par l' equation de Korteweg-de Vries. L'objet de ce travail est de proposer un probl eme mixte bien pos e lorsque le do-maine spatial est born e. Plus pr ecis ement nous etablissons l'existence de solutions locales en temps pour des donn ees ini...

Abstract: This study focus on the solution of the generalized Korteweg and de Vries (KdV) by using homotopy perturbation method (HPM). The HPM has the capabilities to bereave the complicated differential equation models to number of simple iterative models once the effective initial guess satisfying the boundary conditions is made and leads to generic solutions in addition to their rapid conver...

We develop and compare some geometric integrators for the Korteweg-de Vries equation, especially with regard to their robustness for large steps in space and time, ∆x and ∆t, and over long times. A standard, semi-explicit, symplectic finite difference scheme is found to be fast and robust. However, in some parameter regimes such schemes are susceptible to developing small wiggles. At the same i...