A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves
The Korteweg-de Vries equation and the Boussinesq equation have enjoyed the pride of place in almost all investigations of nonlinear fluid motion. Inspite of their importance neither of these equations have been rigourously derived starting from the equations governing the motion of inviscid irrotational fluids. The present authors have attempted to fill this void in the case of two dimensions. Clearly there are two parameters in the problem (1) the ratio of mean water depth to the wave length and (2) the ratio of the amplitude to the mean depth. There are two independent limiting cases to consider (1) shallow water approximation and (2) long wave approximations. Using a conformal mapping of the domain occupied by the fluid and using a scale of Banach spaces of analytic functions, they show that for the Cauchy problem, a short time solution exists which can be approximated uniformly by the corresponding solutions of the Boussinesq and KdV equations. This paper marks an important contribution to rigorous fluid dynamics.
scientific article; zbMATH DE number 4010034