Fourth-order ordinary differential equation obtained by similarity reduction of the modifed Sawada-Kotera equation

TL;DR

This paper investigates a specific fourth-order ODE derived from the modified Sawada-Kotera equation, revealing its polynomial Hamiltonian structure, symmetries, and Bäcklund transformations, contributing new insights into its integrability properties.

Contribution

It introduces a polynomial Hamiltonian form of the ODE, explores its symmetry and holomorphy conditions, and establishes affine Weyl group symmetry in a higher-dimensional symmetric form.

Findings

The ODE can be transformed into a polynomial Hamiltonian system.

The system admits affine Weyl group symmetry of type A2^{(2)}.

A symmetric form in five dimensions is constructed for the Hamiltonian system.

Abstract

We study a one-parameter family of the fourth-order ordinary differential equations obtained by similarity reduction of the modifed Sawada-Kotera equation. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four. We make this polynomial Hamiltonian from the viewpoint of accessible singularity and local index. We also give its symmetry and holomorphy conditions. These properties are new. Moreover, we introduce a symmetric form in dimension five for this Hamiltonian system by taking the two invariant divisors as the dependent variables. Thanks to the symmetric form, we show that this system admits the affine Weyl group symmetry of type $A_2^{(2)}$ as the group of its B{\"a}cklund transformations.