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

Abstract

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.

Solvability of the Model RH Problem for \(\Phi _\sigma \)

We now establish the existence of the solution to the RH problem for $\Phi _ {\sigma }$; to this end we rely on the following vanishing lemma, which shows that there are no nontrivial solutions to the homogeneous version of the RH problem for $\Phi _ {\sigma }$.

Lemma A.1

(Vanishing lemma). Let H be a $2\times 2$ matrix function satisfying the following conditions.

• (a) H is analytic in ${\mathbb {C}}\setminus {\mathbb {R}}$.

• (b) H has boundary values $H_{\pm }$ on ${\mathbb {R}}$ which are $L^2$ on any compact real set, and which are continuous on the real line except at the points $r_1, \ldots ,r_k$, and they are related by

  $$\begin{aligned} H_{+}\left( z\right) =H_{-}\left( z\right) \begin{pmatrix}1&{}1- {\sigma }\left( z\right) \\ 0&{}1\end{pmatrix}. \end{aligned}$$

  (7.27)

• (c) For $|\arg z|<\pi$, we have uniformly

  $$\begin{aligned} H\left( z\right) = \mathcal O\left( |z|^{-3/4}\right) \mathrm{e}^{ x z^{1/2}\sigma _3} \; \text {as} \; z\rightarrow \infty . \end{aligned}$$

  (7.28)

Then $H\left( z\right) =0$ identically in z.

Proof

The proof follows standard arguments, see for example [24]. Set

$$\begin{aligned} M\left( z\right) =H\left( z\right) \begin{pmatrix} 0 &{} -1 \\ 1 &{} 0 \end{pmatrix}H^\dagger \left( {{\overline{z}}}\right) , \end{aligned}$$

where $H^\dagger$ denotes the conjugate transpose of H. Then $M\left( z\right)$ is an analytic function of $z\in {\mathbb {C}}\setminus {\mathbb {R}}$, and as $z\rightarrow \infty$ we have, from (A.2), $M\left( z\right) ={\mathcal {O}}\left( |z|^{-3/2}\right)$. Therefore by Cauchy’s theorem we have

$$\begin{aligned} \int _{\mathbb {R}}M_+\left( z\right) \mathrm {d}z=\int _{\mathbb {R}}H_+\left( z\right) \begin{pmatrix} 0 &{} -1 \\ 1 &{} 0 \end{pmatrix}H_-^\dagger \left( z\right) \mathrm {d}z=0 \end{aligned}$$

implying

$$\begin{aligned} \int _{-\infty }^{+\infty } H_-\left( z\right) \begin{pmatrix} 1 &{} 1- {\sigma }\left( z\right) \\ 0 &{} 1 \end{pmatrix}\begin{pmatrix} 0 &{} -1 \\ 1 &{} 0 \end{pmatrix} H_-^\dagger \left( z\right) \mathrm {d}z =\int _{-\infty }^{+\infty } H_-\left( z\right) \begin{pmatrix} 1- {\sigma }\left( z\right) &{} -1 \\ 1 &{} 0 \end{pmatrix}H_-^\dagger \left( z\right) \mathrm {d}z=0.\nonumber \\ \end{aligned}$$

(A.1)

Adding (A.3) and its adjoint we obtain

$$\begin{aligned} \int _{-\infty }^{+\infty } H_-\left( z\right) \begin{pmatrix} 1- {\sigma }\left( z\right) &{} 0 \\ 0 &{} 0 \end{pmatrix}H_-^\dagger \left( z\right) \mathrm {d}z \\ =\int _{-\infty }^{+\infty }\left( 1- {\sigma }\left( z\right) \right) \begin{pmatrix} |H_{11,-}\left( z\right) |^2 &{} H_{11,-}\left( z\right) {\overline{H}}_{21,-}\left( z\right) \\ {\overline{H}}_{11,-}\left( z\right) H_{21,-}\left( z\right) &{} |H_{21,-}\left( z\right) |^2 \end{pmatrix}\mathrm {d}z=0, \end{aligned}$$

implying $\left( 1- {\sigma }\left( z\right) \right) H_{11,-}\left( z\right) =0$ and $\left( 1- {\sigma }\left( z\right) \right) H_{21,-}\left( z\right) =0$ identically for z on the real line; it follows that

$$\begin{aligned} H_+\left( z\right) -H_-\left( z\right) =H_-\left( z\right) \begin{pmatrix} 0 &{} 1- {\sigma }\left( z\right) \\ 0 &{} 0\end{pmatrix}=\begin{pmatrix} 0 &{} \left( 1- {\sigma }\left( z\right) \right) H_{11,-}\left( z\right) \\ 0 &{} \left( 1- {\sigma }\left( z\right) \right) H_{21,-}\left( z\right) \end{pmatrix}=0,\qquad z\in {\mathbb {R}}, \end{aligned}$$

and so, by the Schwarz reflection principle, $H\left( z\right)$ is analytic throughout the whole complex plane (possible poles at $r_1,\dots ,r_k$ are ruled out by the $L^2$ condition on the boundary values of H). Therefore $H_{11}\left( z\right)$ and $H_{21}\left( z\right)$ vanish identically, because they are entire and they vanish along some nonzero interval of ${\mathbb {R}}$. It follows from this together with the jump condition (A.1) that $H_{12}\left( z\right)$ and $H_{22}\left( z\right)$ are entire functions, which by (A.2) behave as ${\mathcal {O}}\left( z^{-3/4}{\mathrm {e}}^{-xz^{1/2}}\right)$ when $z\rightarrow \infty$ uniformly in all directions of the complex plane; because of our choice of branch for the square root, one concludes from the Liouville theorem that also $H_{12}\left( z\right)$ and $H_{22}\left( z\right)$ vanish identically. $\square$

Finally we follow the standard argument to infer that the solution to the RH problem for $\Phi _ {\sigma }$ exists; indeed, it is known that $\Phi _ {\sigma }$ exists if and only if a certain Fredholm operator of index zero is invertible, and it is easy to see that the kernel of this operator is nontrivial if and only there exists a matrix H with the properties listed in Lemma A.1. We refer to [34, App. A] for a detailed explanation of this argument; in this respect we point out that the RH problem for $\Phi _ {\sigma }$ can be transformed by elementary, though lengthy, transformations into an umbilical RH problem in the sense of [34].