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 '=α g 2 +βζ g + γ transformed to the fourth Painleve equation and several well-known linear ordinary differential equations, or g '=α g 2 + β g + γζ to the second Painleve equation and a few linear equations.
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