Dispersive Schemes for the Critical Korteweg–de Vries Equation

Abstract

In this paper we study semi-discrete finite difference schemes for the critical Korteweg–de Vries equation (cKdV, which is gKdV for k = 4). We prove that the solutions of the discretized equation (using a two grid algorithm) satisfy dispersive estimates uniformly with respect to the discretization parameter. This implies convergence in a weak sense of the discrete solutions to the solution of the Cauchy problem even for rough L2(ℝ) initial data. We also prove a scattering result for the discrete equation, and show that the discrete scattering function converges to the continuous one. Finally rates of convergence are obtained for the approximation of a semi-linear equation with initial data in Hs, s > 0, yet a similar result remains open for the quasilinear cKdV equation. Our analysis relies essentially on the discrete Fourier transform and standard harmonic analysis on the real line.

The Dispersion property for Schrödinger equations

“…In this paper we will not discuss the possible numerical approximation of the NSE as in [17,18,2,23] where there is a parameter h, the mesh size, that it is going to zero.…”

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

“…where K : (0, ∞) → (0, ∞) is a smooth function, see Audiard [4], [3], Benzoni-Gavage et al [6], [5].…”

“…In the latter case, the equations (1.1 -1.3), by using the Madelung transformations, may be formally seen as a description of the evolution of the momenta 4) where the wave function ψ, in the case α = 0 and ̺ = 0, is a solution of the following Schrödinger-Poisson system:…”

Well/ill posedness for the Euler-Korteweg-Poisson system and related problems

“…where K : (0, ∞) → (0, ∞) is a smooth function, see Audiard [4], [3], Benzoni-Gavage et al [6], [5].…”