On the zeros of the Pearcey integral and a Rayleigh-type equation

Abstract

In this work we find a sequence of functions $f_n$ at which the integral

$$\begin{aligned} v(t,x)=\int _{-\infty }^{\infty }e^{i\lambda x-\lambda ^2t/2-\lambda ^4/4}\mathrm{d}\lambda \end{aligned}$$

(1)

is identically zero for all $t\ge 0$, that is

$$\begin{aligned} v(t,f_n(t))=0\qquad \forall t\ge 0. \end{aligned}$$

The function v, after proper change of variables and rotation of the path of integration, is known as the Pearcey integral or Pearcey function, indistinctly. We also show that each $f_n$ is expressed in terms of a second order non-linear ODE, which turns out to be of the Rayleigh-type. Furthermore, the initial conditions which uniquely determine each $f_n$, depend on the zeros of an Lévy stable function of order 4 defined as

$$\begin{aligned} \phi (x)=\int _{-\infty }^{\infty }e^{i\lambda x-\lambda ^4/4}\mathrm{d}\lambda . \end{aligned}$$

As a byproduct of these facts, we develop a methodology to find a class of functions which solve the moving boundary problem of the heat equation. To this end, we make use of generalized Airy functions, which in some particular cases fall within the category of functions with infinitely many real zeros, studied by Pólya.