论文标题
Isomonodromy方法的理性painleve-IV溶液的大型渐近学
Large-degree asymptotics of rational Painleve-IV solutions by the isomonodromy method
论文作者
论文摘要
Painleve-IV方程具有两个由广义的Hermite多项式和广义冈本多项式产生的有理解的家族。我们应用异构方法通过两个相关的riemann-hilbert问题来表示所有这些有理解，每种问题涉及两个与Painleve-IV方程中两个参数相关的整数值参数。然后，我们使用最陡峭的方法来分析至少一个参数很大的极限。我们的分析为形式的渐近论参数提供了严格的理由，这些论点表明，在大型参数的一般解决方案中，具有大量参数的painleve-IV解决方案是代数函数或椭圆函数。此外，结果表明，椭圆形近似是曲线矩形的结合，并且在广义的冈本有理溶液的情况下，四个曲线三角形的三角形都与矩形具有边缘。代数近似在互补的无界域中有效。我们将杆和零位置的理论预测与从生成多项式获得的实际极线和零的数值图进行了比较，并找到了极好的一致性。
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painleve-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement.
