Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada – Kotera and Kupershmidt Equations

Self-similar reductions of the Sawada – Kotera and Kupershmidt equations are studied. Results of Painlevé's test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada – Kotera and Kupershmidt equations expressed via the first Painlevé equation are presented. Exact solutions of the Sawada – Kotera and Kupershmidt equations by means of general solution for the first member of $K_2$ hierarchy are given. Special polynomials for expressions of rational solutions for the equations considered are introduced. The differentialdifference equations for finding special polynomials corresponding to the Sawada – Kotera and Kupershmidt equations are found. Nonlinear differential equations of sixth order for special polynomials associated with the Sawada – Kotera and Kupershmidt equations are obtained. Lax pairs for nonlinear differential equations with special polynomials are presented. Rational solutions of the self-similar reductions for the Sawada – Kotera and Kupershmidt equations are given.