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.)