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 \(t\rightarrow 0\), and hence would be well-defined as solutions of the cylindrical Korteweg–de Vries equation. We provide uniform asymptotics in x as \(t\rightarrow 0\); 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.

我们研究了 Korteweg-de Vries 方程的一系列无界解，它们可以构造为变形的 Airy 核 Fredholm 行列式的对数导数，并且连接到第二个 Painlevé 方程的积分微分版本。Korteweg–de Vries 解的初始数据对于\(x>0\)是明确定义的，但不是对于\(x<0\)，其中解的行为类似于\(\frac{x}{2t}\ )作为\(t\rightarrow 0\)，因此可以很好地定义为圆柱 Korteweg-de Vries 方程的解。我们提供x 中的均匀渐近线为\(t\rightarrow 0\)；对于\(x>0\)它们涉及 Painlevé V 方程的积分微分模拟。我们结果的一个特例对 Kardar-Parisi-Zhang 方程的窄楔形解的尾部产生了改进的估计。