Higher Order Modulation Equations for a Boussinesq Equation

Abstract

In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. We also present the results of numerical experiments which show that the error estimates we derive are essentially optimal.

References in corpus (1)

• Inversion of the linearized Korteweg-deVries equation at the multi-soliton solutions

Cited by in corpus (6)

• Ill-posedness of degenerate dispersive equations
• Weakly nonlinear extension of d'Alembert's formula
• D'Alembert-type solution of the Cauchy problem for a Boussinesq-Klein-Gordon equation
• Corrections to the KdV approximation for water waves
• On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations
• Validity of the weakly-nonlinear solution of the Cauchy problem for the Boussinesq-Ostrovsky equation