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Since these equations are approaching three hundred years old I'm pretty sure someone must have solved them somewhere before. However I have not been able to find any text-books or papers that actually show me how to solve these equations. So I'm wondering if anyone here know where I can find an approach to the analytical solutions, or who simply know how I should attack a problem set like this?

Here are the equations:

\begin{align}

\frac{\partial\rho}{\partial t} + \frac{\partial}{\partial x}(\rho v) &= 0 \\

\frac{\partial}{\partial t}(\rho v) + \frac{\partial}{\partial x}(\rho v^2) +\frac{\partial p}{\partial x} &= - \rho\frac{\partial \phi}{\partial x} \\

\frac{\partial^2\phi}{\partial x^2}& = 4\pi G\rho

\end{align}

Were $t$ and $x$ are the space and time coordinates, $\rho = \rho(x,t)$ is a density, $v=v(x,t)$ is a velocity, $p=p(\rho)$ is the pressure related to the density with a constant $p=w\rho$ and $\phi=\phi(x,t)$ is a gravitational potential. The physical variables isn't really important here, as I am more curious as to the mathematical approach to find one analytical solution.

A general analytical solution probably doesn't exist for this set, but I believe I can give some initial conditions:

\begin{align}

\rho_0 = 1,\quad v_0 = sin(2\pi x), \quad p_0= 10^{-5}

\end{align}

Anyone got any sources or tips for solving this? Having multiple variables ($\rho$, $v$) as well as several partial derviatives ($\partial x, \partial t$) in the same equations is what is putting me off.

Thanks in advance for any help.

P.S.

Wasn't sure of where to put this thread, but decided for Math as the physics is kinda irrelevant at this point I think.

Cheers

EDIT: Oh, I see writing TeX in the text is not how it is in LaTeX on this forum, how should I do it?

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# Analytical solution of the Euler equations in 1D?

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