Irreducibility and co-primeness as an integrability criterion for discrete equations
Irreducibility and co-primeness as an integrability criterion for discrete equations
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不可约性和互质性作为离散方程的可积性准则
DOI：
10.1088/1751-8113/47/46/465204
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
間瀬崇史
中科院分区：
文献类型：
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作者：
江川達郎;大野善隆;後藤亜由美;横山真吾;生田旭洋;鈴木美穂;林達也;後藤勝正;間瀬崇史
研究了离散可积方程和不可积方程的Laurent性质、不可约性和余素性。作为比较，我们首先研究了一个与Somos-4序列有关的离散可积方程和一个不可积方程。我们证明了不可约和余素性的条件只在可积情况下成立。接下来，我们推广了以前关于离散Korteweg-de Vries(DKdV)方程奇性的结果。在我们的前一篇论文中(Kanki等人2014 J.Phys.答：数学。西奥。47 065201)，我们利用洛朗性质和双线性dKdV方程中项的不可约性和余素性描述了奇性约束检验(可积性准则之一)，其中我们只考虑了简化的边界条件。为了获得dKdV方程的双线性形式和非线性形式之间的简单(单项)关系，需要这种限制。本文利用Laurent环的局部化和轴的互换，证明了非线性dKdV方程在一般初始条件和边界条件下项的余素性。我们断言项的余素性可以用作一个新的可积性判据，它是对离散方程中奇点限制的数学重新解释。
We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg–de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor. 47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations.