Derivation of Bessel functions I am writing a summary on a work on Fluid Dynamics that develops irrotational flow states that appear to interact amongst each other according to the equations of Electromagnetism http://arxiv.org/abs/1301.7540
So it begins with Euler Equations of inviscid compressible fluid. Apply some constraints and then find a solution. The solution is a Bessel function:
$$\left.\begin{array}{rcl}
 \dfrac{\partial \mathbf{u}}{\partial t} \;+\; (\mathbf{u}.\nabla)\mathbf{u}& \; = \; -\: \dfrac{1}{\rho} \nabla P  \\\
 \rho(\mathbf{x}, t)& \; \ll \; 1  &  
\end{array}\right\rbrace$$
$$\Rightarrow \xi =\: \psi_o(t)\: R_{mn}(\mathbf{x}) \;\;: \\\
\begin{cases}
\mathrm{Re}(\xi) &\overset{\underset{\mathrm{def}}{}}{=}\: \dfrac{\rho}{\rho_o} - 1 \\\
 \psi_o &\overset{\underset{\mathrm{def}}{}}{=}\; A \: e^{-i\omega_ot} \\\
\displaystyle R_{mn} &\overset{\underset{\mathrm{def}}{}}{=}\;  \int_{0}^{2\pi} e^{-i(m{\theta}'\,-\,n\phi )}j_m(\kappa_r\sigma)\kappa_rR_o \mathbf{d} \phi
\end{cases}
$$
My goal is to do a step-by-step proof of his derivation and learn somethings about such system. Later I would like to derive step-by-step how two such systems interact with each other, if possible. The article is rather dry on the derivations as it assumes these are rather uninteresting and unremarkable.

Update 1: So far I have found online derivations to the Euler equation and a very attractive derivation of Bessel functions with gorgeous physical insights to it:

http://galileo.phys.virginia.edu/classes/311/notes/fluids1/fluids11/node10.html
http://physics.ucsc.edu/~josh/116C.07/bessel/node1.html
I can't apply the derivation of Bessel directly because it starts from the equation:
$\nabla^2\mathbf{u}(x, y, z) = 0$ . I don't know how to relate that to the Euler equation of the form $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}.\nabla)\mathbf{u}  = - \frac{1}{\rho} \nabla P$. Does someone know how the two relate?

Update 2:
Carlo Beenakker pointed out that the target solution ignores the effects of the convective term of Euler equation: $(\mathbf{u} \cdot \nabla)\mathbf{u}$

That relates to the reference article in that the author makes the assumption of "low amplitude", meaning $\mathbf{u} \ll 1$
Carlo Beenakker has also given a full answer I am still studying. I hope it is complete but I would appreciate anyone helping. I should take a couple of days.

Background: I am not a professional mathematician or physicist. I know the proper way to do this would take a couple of semesters and do the proper college courses on differential equations, with much calculus background, which I don't have. As my interest is mostly only on this specific set of equations and I don't have a tutor or teacher to help me I would need some points on what would be the fastest way to finish this such that the math is rigorous.

I hope some of you have any interest for this curious approach too. Thank you for helping.
PS: Don't mind the article talks about Quantum Mechanics. Im not interested in that.
(I eliminated references to QM to avoid misundertandings.)
 A: I'll make an attempt at providing the steps you are seeking to go "from Euler equation to Bessel function".
You start from the Euler equation, describing conservation of momentum,
$$\rho\frac{\partial \vec{u}}{\partial t}+\rho\vec{u}\cdot\nabla\vec{u}=-\nabla p$$
and the continuity equation, describing conservation of mass,
$$\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho \vec{u})$$
These are nonlinear equations, to make them tractable you'll want to linearize them, both in the velocity $\vec{u}$ and in the deviations $\delta\rho=\rho-\rho_0$ of the density from the uniform density $\rho_0$. This approximation throws away lots of interesting physics (shock waves, turbulence,...), but without it no simple solution exists.
The linearized equations read
$$\rho_0\frac{\partial \vec{u}}{\partial t}=-\nabla p$$
$$\frac{\partial\delta\rho}{\partial t}=-\rho_0\nabla\cdot\vec{u}$$
We may also assume a linear relation $p=p_0+C^2\delta\rho$ between the pressure $p$ and the density variations. (This is a socalled adiabatic equation of state, the coefficient $C^2$ must be positive for mechanical stability.) We define $\xi=\delta\rho/\rho_0$, take the divergence of the first equation and the time derivative of the second equation,
$$\nabla\cdot\frac{\partial \vec{u}}{\partial t}=-C^2\nabla^2 \xi$$
$$\frac{\partial^2\xi}{\partial t^2}=-\frac{\partial}{\partial t}\nabla\cdot\vec{u}$$
Finally, we substitute the first equation into the second one, exchanging the order of differentiation with respect to time and space, to arrive at a wave equation for $\xi$,
$$\frac{\partial^2\xi}{\partial t^2}=C^2\nabla^2 \xi$$
The quantity $C>0$ represents the speed of sound.
We seek a solution of this equation that is a harmonic function of time, so it oscillates with frequency $\omega$. Rather than working with sines or cosines, it is more convenient to use a complex notation, writing
$$\xi(\vec{r},t)={\rm Re}\;e^{-i\omega t}f(\vec{r})$$
The complex function $f$ satisfies the Poisson equation,
$$C^2\nabla^2 f=-\omega^2 f$$
Let's seek a solution with cylindrical symmetry, so $f(R)$ depends only on the radial coordinate $R=\sqrt{x^2+y^2}$. The Poisson equation in cylindrical coordinates takes the form
$$\frac{d^2}{dR^2}f(R)+\frac{1}{R}\frac{d}{dR}f(R)=-(\omega/C)^2f$$
The solution is a Bessel function
$$f(R)={\rm constant}\times J_0(\omega R/C)$$
The full solution thus becomes
$$\delta\rho/\rho_0=A\cos(\omega t+B)J_0(\omega R/C)$$
where $A$ and $B$ are arbitrary coeffients.
And we're done :)
