Large-Degree Asymptotics of Rational Painlevé-IV Functions Associated to Generalized Hermite Polynomials
Large-Degree Asymptotics of Rational Painlevé-IV Functions Associated to Generalized Hermite Polynomials
复制标题
与广义 Hermite 多项式相关的有理 Painlevé-IV 函数的大度渐近
DOI：
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发表时间：
2017
影响因子：
1
通讯作者：
R. Buckingham
中科院分区：
文献类型：
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作者：
R. Buckingham
Painlevé-IV方程有三族由广义Hermite多项式生成的有理数解。每个族由两个正整数$m$和$n$索引。这些函数可应用于非线性波动方程、随机矩阵、流体力学和量子力学。数值研究表明，零点和极点形成了一个变形的矩形网格。经过适当的缩放，零点和极点似乎密密麻麻地将某些曲线矩形填充为$m，n o inty$，其中$r：=m/n$是固定的正实数。推广了Bertola和Bothner[2]研究有理Painlevé-II函数的方法，将广义Hermite有理Painlevé-IV函数表示为某些非Hermite正交多项式.利用Deift-周非线性最速下降法，我们在极限n=n，n，m=rcdon，r，n固定的条件下，对相应的Riemann-Hilbert问题进行了渐近分析。我们得到了边界曲线的一个显式刻画，并确定了与广义Hermite多项式相关的有理Painlevé-IV函数在无极点区域的前导阶渐近展开式。
The Painlevé-IV equation has three families of rational solutions generated by the generalized Hermite polynomials. Each family is indexed by two positive integers $m$ and $n$. These functions have applications to nonlinear wave equations, random matrices, fluid dynamics, and quantum mechanics. Numerical studies suggest the zeros and poles form a deformed $n imes m$ rectangular grid. Properly scaled, the zeros and poles appear to densely fill certain curvilinear rectangles as $m,n o infty $ with $r:=m/n$ a fixed positive real number. Generalizing a method of Bertola and Bothner [2] used to study rational Painlevé-II functions, we express the generalized Hermite rational Painlevé-IV functions in terms of certain non-Hermitian orthogonal polynomials. Using the Deift–Zhou nonlinear steepest-descent method, we asymptotically analyze the associated Riemann–Hilbert problem in the limit $n o infty $ with $m=rcdot n$ for $r$ fixed. We obtain an explicit characterization of the boundary curve and determine the leading-order asymptotic expansion of the rational Painlevé-IV functions associated to generalized Hermite polynomials in the pole-free region.