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

*2020, Volume 25, Number 1, pp. 59-77*

Author(s):

**Kudryashov N. A.**

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.

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