This article concerns the nonlinear Korteweg-de Vries equation with boundary timedelay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system...

— In this paper we give an algorithmic method of deriving the Lax pair for the modified Korteweg-de Vries hierarchy. For each n, the compatibility condition gives the n-th member of the hierarchy, rather than its derivative. A direct consequence of this is that we obtain the isomonodromy problem for the second Painlevé hierarchy, which is derived through a scaling reduction. Résumé (La paire de...

When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon when the time is rational or irrational relative to the length of the interval: the Talbot phenomenon of dispersive quantization and fractalization. The purpose of this paper is to show that these phenome...

In this paper, we find the relationship between the solution of (1+1)-dimensional Korteweg-de Vries (KdV) equation and the solution of (2+1)-dimensional integrable Schwarz-Korteweg-de Vries(SKdV) equation with Möbius transformations, Miura transformation and other transformations. Furthermore, we can obtain the new solution of the SKdV equation in 2+1 dimensions by using the solution of KdV equ...

The cnoidal wave solution of the integrable Korteweg de Vries equation is the most basic of its periodic solutions. Following earlier work where the linear stability of these solutions was established, we prove in this paper that cnoidal waves are (nonlinearly) orbitally stable with respect to so-called subharmonic perturbations: perturbations that are periodic with period any integer multiple ...

Linear and non-linear surface waves on a ferrofluid cylinder surrounding a current-carrying wire are investigated. Suppressing the Rayleigh-Plateau instability of the fluid column by the magnetic field of a sufficiently large current in the wire axis-symmetric surface deformations are shown to propagate without dispersion in the long wavelength limit. Using multiple scale perturbation theory th...