论文标题
$ a_ {2n} $ - painlevé系统的理性解决方案的完整分类
Complete classification of rational solutions of $A_{2n}$-Painlevé systems
论文作者
论文摘要
我们提供了针对第四个Painlevé方程piv及其高阶概括的完整分类和明确表示，称为$ a_ {2n} $ - painlevé或noumi-yamada系统。解决方案的构建利用了Schrödinger操作员的循环敷料链的理论。研究溶液围绕其奇异性的局部扩展，我们发现其lourent膨胀中的某些系数必须消失，这些系数精确地表达了相关电位的琐碎单构成条件。具有二次生长的微不足道单构电位的表征表明，所有有理溶液都可以表示为适当选择的Hermite多项式序列的Wronskian决定因素。主要分类结果指出，$ a_ {2n} $ - painlevé系统的每个合理解决方案都对应于玛雅图的周期，可以用奇怪的彩色整数序列进行索引。最后，我们通过在对称序列和玛雅周期上的对称群体作用的表示形式来基于在种子溶液上应用Bäcklund变换的标准方法建立链接。
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the $A_{2n}$-Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the $A_{2n}$-Painlevé system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Bäcklund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.
