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Non-integrability of the Sasano system of type $A^{(2)}_5$

Tsvetana Stoyanova

TL;DR

The paper investigates the non-integrability of the four-dimensional Sasano system of type $A^{(2)}_5$ by applying the Morales–Ramis–Simó framework to the first and second normal variational equations along a rational seed solution. By computing the differential Galois groups and analyzing Stokes data for irregular singularities, it establishes non-Abelian obstructions to Liouville integrability in the generic parameter regime and in a key special case. The use of Bäcklund transformations then extends these obstructions to all parameter values for which a rational solution exists, yielding the main non-integrability result. Overall, the work advances the integrability classification of higher-order Painlevé-type systems through a rigorous differential Galois-theoretic approach.

Abstract

The Sasano sytem of type $A^{(2)}_5$ is a four-dimensional non-linear system of ordinary differential equations, which has an affine Weyl group of symmetries of type $A^{(2)}_5$. It is also a tipe dependent Hamiltonian system, which can be considered as coupled Painlevé III systems. In this paper, utilizing the Morales-Ramis-Simó theory for integrability of Hamiltonian systems, we prove rigorously that for all values of the parameters, for which the Sasano system of type $A^{(2)}_5$ admits a particular rational solution, it is non-integrable by rational first integrals.

Non-integrability of the Sasano system of type $A^{(2)}_5$

TL;DR

The paper investigates the non-integrability of the four-dimensional Sasano system of type by applying the Morales–Ramis–Simó framework to the first and second normal variational equations along a rational seed solution. By computing the differential Galois groups and analyzing Stokes data for irregular singularities, it establishes non-Abelian obstructions to Liouville integrability in the generic parameter regime and in a key special case. The use of Bäcklund transformations then extends these obstructions to all parameter values for which a rational solution exists, yielding the main non-integrability result. Overall, the work advances the integrability classification of higher-order Painlevé-type systems through a rigorous differential Galois-theoretic approach.

Abstract

The Sasano sytem of type is a four-dimensional non-linear system of ordinary differential equations, which has an affine Weyl group of symmetries of type . It is also a tipe dependent Hamiltonian system, which can be considered as coupled Painlevé III systems. In this paper, utilizing the Morales-Ramis-Simó theory for integrability of Hamiltonian systems, we prove rigorously that for all values of the parameters, for which the Sasano system of type admits a particular rational solution, it is non-integrable by rational first integrals.

Paper Structure

This paper contains 7 sections, 20 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-integrability of Hamiltonian systems and differential Galois theory
  4. The differential Galois group at an irregular singularity
  5. Proof of Theorem \ref{['tkey1']}
  6. Proof of Theorem \ref{['tkey2']}
  7. Proof of the Theorem \ref{['tmain']}

Key Result

Theorem 1.1

(Matsuda) For a rational solution of the Sasano system s, by some Bäcklund transformations, the solution and the parameters can be transformed so that respectively. Furthermore, for the Sasano system s, there exists a rational solution if and only if one of the following occurs:

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 17 more