Corrections to the KdV Approximation for Water Waves

Abstract

Abstract. In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone.

A shallow‐water approximation to the full water wave problem

“…It is also worth mentioning some recent work on the water wave problem, including Lannes' proof [20] of the existence of solutions using the Eulerian coordinates under the assumption of variable-bottom topography in n-dimensional space for any n ≥ 2; Ambrose's result [1] on well-posedness of vortex sheets with surface tension; Wright's [34] work on corrections to the KdV approximation; and Bona, Colin, and Lannes's [6] result on a class of Boussinesq equations as a good approximation to the water wave problem.…”

Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation

“…Although the PDE (1.5) looks very simple, it has a lot of applications, for example, Whitham [14] used it for the modeling of the propagation of long waves in the shallow water equations, see also [15]. We emphasize the fact that the restriction of the solution to Eq.…”

Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

“…The second group of equations, namely (11), (13), and (14), describes the essential interaction dynamics. By pure integration we find Lemma 4.6.…”

“…In the same spirit there are approximation results describing the interaction of KdV-described long waves by higher-order approximations in FPU-lattices [7], in Boussinesq models [12], and in the water wave problem [13]. …”