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Non-Integrability of the Sasano System of Type $D_5^{(1)}$ and Stokes Phenomena

Tsvetana Stoyanova

TL;DR

The paper analyzes the four‑dimensional Sasano system of type $D^{(1)}_5$, a coupled Painlevé‑V‑type system with extended symmetry, to establish non‑integrability by rational first integrals. It applies the Morales‑Ramis‑Simó framework, computing the normal variational equations and their differential Galois groups, and explicitly shows a non‑Abelian obstruction from a higher‑order LNVE together with Stokes data. By exploiting Bäcklund transformations of the extended Weyl group $W(D^{(1)}_5)$, the non‑integrability result is extended to broad parameter orbits, including both non‑integer and parity‑restricted integer cases. The methodology yields explicit analytic invariants (Stokes matrices) and demonstrates that the connected component of the identity in the differential Galois group fails to be Abelian, thus ruling out Liouville–Arnol'd integrability by rational first integrals in the stated regimes. This work strengthens understanding of higher‑order Painlevé systems and showcases a concrete differential Galois obstruction approach with Stokes phenomena for non‑autonomous Hamiltonian systems.

Abstract

In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type $D^{(1)}_l$, $l=4, 5, 6, \dots$. In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group $\widetilde{W}\bigl(D^{(1)}_5\bigr)$ and which we call the Sasano system of type $D^{(1)}_5$. We prove that one family of the Sasano system of type $D^{(1)}_5$ is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result.

Non-Integrability of the Sasano System of Type $D_5^{(1)}$ and Stokes Phenomena

TL;DR

The paper analyzes the four‑dimensional Sasano system of type , a coupled Painlevé‑V‑type system with extended symmetry, to establish non‑integrability by rational first integrals. It applies the Morales‑Ramis‑Simó framework, computing the normal variational equations and their differential Galois groups, and explicitly shows a non‑Abelian obstruction from a higher‑order LNVE together with Stokes data. By exploiting Bäcklund transformations of the extended Weyl group , the non‑integrability result is extended to broad parameter orbits, including both non‑integer and parity‑restricted integer cases. The methodology yields explicit analytic invariants (Stokes matrices) and demonstrates that the connected component of the identity in the differential Galois group fails to be Abelian, thus ruling out Liouville–Arnol'd integrability by rational first integrals in the stated regimes. This work strengthens understanding of higher‑order Painlevé systems and showcases a concrete differential Galois obstruction approach with Stokes phenomena for non‑autonomous Hamiltonian systems.

Abstract

In 2006, Y. Sasano proposed higher-order Painlevé systems, which admit affine Weyl group symmetry of type , . In this paper, we study the integrability of a four-dimensional Painlevé system, which has symmetry under the extended affine Weyl group and which we call the Sasano system of type . We prove that one family of the Sasano system of type is not integrable by rational first integrals. We describe Stokes phenomena relative to a subsystem of the second normal variational equations. This approach allows us to compute in an explicit way the corresponding differential Galois group and therefore to determine whether the connected component of its unit element is not Abelian. Applying the Morales-Ramis-Simó theory, we establish a non-integrable result.
Paper Structure (6 sections, 24 theorems, 49 equations)

This paper contains 6 sections, 24 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-integrability of Hamiltonian systems and differential Galois theory
  4. Differential Galois group of a linear system of ordinary differential equations
  5. Non-integrability for $\boldsymbol{\alpha_1=\alpha_2=\alpha_3=0}$, $\boldsymbol{\alpha_4=1}$, $\boldsymbol{\alpha_0=-\alpha_5}$
  6. Bäcklund transformations and generalization

Key Result

Theorem 1.1

Assume that $\alpha_1=\alpha_2=\alpha_3=0$, $\alpha_4=1$, $\alpha_0=-\alpha_5$, where $\alpha_5$ is arbitrary. Then the Sasano system s is not integrable in the Liouville--Arnold sense by rational first integrals.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Morales--Ramis
  • Theorem 2.2: Morales--Ramis--Simó
  • Definition 2.3
  • Theorem 2.4: Schlesinger
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 22 more