Painlevé expansions, truncated at various stages, are constructed for the conditionally-Painlevé Benjamin–Bona–Mahoney (BBM), the modified Benjamin–Bona–Mahoney (MBBM), and the symmetric regularized long wave (SRLW) equations. Expansions truncated at the constant term lead to auto-Bäcklund transformations between two solutions of all three equations. Special solutions of the various equations, including a solitary-wave solution for the MBBM equation, and new one-parameter families of traveling-wave solutions for the BBM and SRLW equations are obtained using the truncated Painlevé expansions.
