Stationary periodic solutions of the two-dimensional Gross-Pitaevskii equation are obtained and analyzed for different parameter values in the context of the problem of a supersonic flow of a Bose-Einstein condensate past an obstacle. The asymptotic connections with the corresponding periodic solutions of the Korteweg-de Vries and nonlinear Schrödinger equations are studied and typical spatial ...

We provide a new analytical approach to operator splitting for equations of the type ut = Au + B(u) where A is a linear operator and B is quadratic. A particular example is the Korteweg–de Vries (KdV) equation ut−uux +uxxx = 0. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.

This article presents a formulation of the time-fractional generalized Korteweg-de Vries (KdV) equation using the Euler-Lagrange variational technique in the Riemann-Liouville derivative sense. It finds an approximate solitary wave solution, and shows that He’s variational iteration method is an efficient technique in finding the solution.

This Brief Report is devoted to the study of the solitary surface wave rotating in the azimuthal direction, arising during water drainage from a cylindrical reservoir, when shallow flow conditions are reached. The linear dependence between the wave speed and its amplitude is shown to be similar to that expected from the classical Korteweg-de Vries equation.

We present in this article all Hamiltonian systems in E(2) that are separable in cartesian coordinates and that admit a third-order integral, both in quantum and in classical mechanics. Many of these superintegrable systems are new, and it is seen that there exists a relation between quantum superintegrable potentials, invariant solutions of the Korteweg-De Vries equation and the Painlevé trans...

In this paper we study the long time dynamics of solutions of Initial and Boundary Value Problem (IBVP) of the Korteweg-de Vries (hence-forth KdV) equation posed on a bounded domain: (*) . 0 , 0 ) , 1 ( , 0 ) , 1 ( ), ( ) , 0 ( , 0 , 1 0 ), ( ) 0 , ( , 0    > = = = > < < = = + + + t t u t u t h t u t x x x u u uu u u

We prove local well-posedness of the initial-boundary value problem for the Korteweg-de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.

The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains was established in [15] by means of Hilbert Uniqueness Method. The aim of these notes is to illustrate this approach by numerical simulations. 256 ESAIM: Proc., Vol. 4, 1998, 255-267

We extend the dispersion-velocity particle method that we recently introduced to advection models in which the velocity does not depend linearly on the solution or its derivatives. An example is the Korteweg de Vries (KdV) equation for which we derive a particle method and demonstrate numerically how it captures soliton–soliton interactions.

Explicit motion planning of a linearized Korteweg-de Vries equation with boundary control is achieved. The control is obtained through a “parametrization” of the trajectories of the system which is a generalization of the Brunovsky decomposition for finite-dimensional linear systems.Copyright c © 2002 IFAC