Airy Kernel Determinant Solutions to the KdV Equation and Integro-Differential Painlevé Equations

We study a family of unbounded solutions to the Korteweg–de Vries equation which can be constructed as log-derivatives deformed Airy kernel Fredholm determinants, and are connected an integro-differential version second Painlevé equation. The initial data well-defined for $$x>0$$ , but not $$x<0$$ where behave like $$\frac{x}{2t}$$ $$t\rightarrow 0$$ hence would cylindrical provide uniform asymptotics in x ; they involve analogue V A special case our results yields improved estimates tails narrow wedge solution Kardar–Parisi–Zhang