The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure i...

Abstract. We consider a Korteweg-de Vries equation perturbed by a noise term on a bounded interval with periodic boundary conditions. The noise is additive, white in time and “almost white in space”. We get a local existence and uniqueness result for the solutions of this equation. In order to obtain the result, we use the precise regularity of the Brownian motion in Besov spaces, and the metho...

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u = 2(u)x, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is uniformly globally well-posed in Hs (s ≥ 1) for all ǫ ∈ (0, 1]. Moreover, we prove that for any s ≥ 1 and T > 0, its solution converges in C([0, T ]; Hs) to that of the MKdV equation if ǫ tends to 0.

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...

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating ...