Similarity Reductions and Painlevé Properties for the Kupershmidt Equations

Abstract

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 Ablowitz-Ramani-Segur algorithm.