论文标题
kDV方程和无数差异painlevé方程的通风内核决定符解决方案
Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
论文作者
论文摘要
我们研究了Korteweg-de Vries方程的无限解决方案家族，该家族可以构建为变形的通风核弗雷德霍尔姆决定因素的对数衍生物，并与第二个PANELEVé方程式的无数分化版本相连。 Korteweg-de Vries解决方案的初始数据定义为$ x> 0 $，但不能以$ x <0 $为$ x <0 $，在这里，该解决方案的行为就像$ \ frac {x} {2t} $作为$ t \ to $ t \ to 0 $，而此外，此解决方案将被很好地定义为callindical korteweg-de korteweg-de vries iess equient。我们以$ x $为$ t \ to 0 $提供统一的渐近物；对于$ x> 0 $，它们涉及painlevév方程式的全差异类似物。我们的结果的一种特殊情况可以改善对kardar-parisi-zhang方程的窄楔解决方案的{尾巴}的估计。
We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation.
