Airy heat bullets

Abstract

New localized structured solutions for the three-dimensional linear heat (diffusion) equation are presented. These new solutions are written in terms of Airy functions. They are constructed as wave packet-like structures formed by a superposition of Bessel functions through the introduction of spectral functions. These diffusive solutions accelerate along their propagation direction, while in the plane orthogonal to it, they retain their confined structure. These heat (diffusion) densities retain a complete localized form in space as they propagate, and may be considered the heat analogue of Airy light bullets.

Appendix

The asymptotic behavior of Airy function is [51]

$$\begin{aligned} {\text{ Ai }}(x)\sim \frac{\exp (-2x^{3/2}/3)}{2\sqrt{\pi }x^{1/4}}\left( \alpha _0+\frac{\alpha _1}{x^{3/2} }\right) \, , \end{aligned}$$

(A.1)

with $\alpha _0=\Gamma (1/6)\Gamma (5/6)/(2\sqrt{\pi })$ and $\alpha _1=-3\, \Gamma (7/6)\Gamma (11/6)/(8\sqrt{\pi })$. Similarly, the asymptotic behavior of derivative of Airy function is

$$\begin{aligned} {\text{ Ai }}'(x)\sim -\frac{x^{1/4}\exp (-2x^{3/2}/3)}{2\sqrt{\pi }}\left( \alpha _0-\frac{7\alpha _1}{5x^{3/2} }\right) \, . \end{aligned}$$

(A.2)

Using these results in Eq. (11), we are able to find the approximated solution (12).