Similarity Reductions and Painlev6 Properties for the Kupershmidt Equations

Five types of similarity reductions of the Kupershmidt equations which admit a tri-Hamiltonian structure are found by a direct method. Two types of reduction equations which are Painlevé Ⅱ and IV types are coincident with those obtained by classical Lie approach. Both algebraic and logarithmic branch points for time t can be entered into the solutions of Kupershmidt equations. The integrability of the Kupershmidt equations is re-examined by the singularity analysis using the Weiss-Kruskal approach and the A blowitz-Ramani-Segur algorithm.