Linearization of the Classical Boussinesq and Related Equations

Abstract

The classical Boussinesq equation is connected with the higher order water wave equation introduced by Kaup through a simple transformation. It is shown that similarity solutions of the two equations satisfy a special Riccati equation g′=αg²+βζg+γ transformed to the fourth Painlevé equation and several well-known linear ordinary differential equations, or g′=αg²+βg+γζ to the second Painlevé equation and a few linear equations.