A family of integrable and non-integrable difference equations arising from cluster algebras (Mathematical structures of integrable systems and their applications)

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説明

The one-parameter family of second order nonlinear difference equations each of which is given by xn-1xnxn+1 = xn-1 + (xn)β-1 + xn+1 (β ∈ N) is explored. Since the equation above is arising from seed mutations of a rank 2 cluster algebra, its solution is periodic only when β ≤ 3. In order to evaluate the dynamics with β ≥ 4, the algebraic entropy of the birational map equivalent to the difference equation is investigated; it vanishes when β = 4 but is positive when β ≥ 5. This fact suggests that the difference equation with β ≤ 4 is integrable but that with β ≥ 5 is not. It is moreover shown that the difference equation with β ≥ 4 fails the singularity confinement test. This fact is consistent with linearizability of the equation with β = 4 and reinforces non-integrability of the equation with β ≥ 5.

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詳細情報 詳細情報について

  • CRID
    1050569159935480320
  • NII論文ID
    120006950568
  • NII書誌ID
    AA12196120
  • ISSN
    18816193
  • HANDLE
    2433/260632
  • 本文言語コード
    en
  • 資料種別
    departmental bulletin paper
  • データソース種別
    • IRDB
    • CiNii Articles

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