Summary: We discuss similarity reductions and Painlevé analysis for the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations, both of which arise in several physical applications including shallow water waves. Both equations are thought to non-integrable (i.e. not solvable by inverse scattering) since numerical studies show that the interaction of solitary waves is inelastic.
In particular, we determine some new similarity reductions of the symmetric regularised long wave equation. These new similarity reductions are not obtainable using the classical Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method which involves no group theoretical techniques.
It is shown that every similarity reduction of both the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations obtained using the classical Lie group method reduces the associated partial differential equation to an ordinary differential equation of Painlevé type; whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of Painlevé type.
It is also shown that neither the symmetric regularised long wave equation nor the modified Benjamin-Bona-Mahoney equation possesses the Painlevé property for partial differential equations as defined by Weiss et al.