论文标题
关于$ q $-painlevévi及其riemann-hilbert问题的单片歧管
On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem
论文作者
论文摘要
我们研究了第六$ q $ -Differencepainlevé方程（$ q {\ textrm {p} _ {\ textrm {vi}}} $）通过其关联的Riemann-Hilbert问题（RHP），并表明RHP始终可用于不可降低的单膜模型数据。这使我们能够用nodromy歧管为通用参数值确定$ q {\ textrm {p} _ {\ textrm {\ textrm {\ textrm {\ textrm {\ textrm {\ textrm {\ textrm {\ textrm {p} _ {\ textrm {p} _ {\ textrm {p} _ {\ textrm {p} _ {\ textrm {p} _ {\ textrm {p} _ {我们明确推断出这种歧管，并表明当它不包含可简化的单曲率时，它是平滑而仿生的代数表面。此外，我们描述了可简化单型数据数据的RHP，并表明，当可解决方案时，其解决方案将以某些正交多项式明确给出，这些正交多项式产生了$ Q {\ textrm {p} {p} _ {\ textrm {vi}}} $的特殊功能解决方案。
We study the sixth $q$-difference Painlevé equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify the solution space of $q{\textrm{P}_{\textrm{VI}}}$ with a monodromy manifold for generic parameter values. We deduce this manifold explicitly and show it is a smooth and affine algebraic surface when it does not contain reducible monodromy. Furthermore, we describe the RHP for reducible monodromy data and show that, when solvable, its solution is given explicitly in terms of certain orthogonal polynomials yielding special function solutions of $q{\textrm{P}_{\textrm{VI}}}$.
