We consider a model describing the flow of a gas in a long tube. Solve the system by using the characteristics: \begin{aligned} p_t + vp_x + c^2\rho v_x &= 0 \\ \rho (v_t v v_x) + p_x &= 0 \end{aligned} Here, $p$, $v$ and $\rho$ are the gas pressure, axial velocity and density and $c$ is the speed of the sound in the gas, $x$ denotes the position and $t$ is the time.
My thoughts: To solve this system, we shall conclude a system as : $$ (p_t+ v_t)+ (p_x+ v_x)=0 $$ but how can I reduces the above system to a system with variable or constant coefficients I can solve, or If I find the value of density of the gas, axial velocity, and speed of the in the gas, I may solve, but how can I find them?
There is an obvious mistake in the problem statement, which should be \begin{aligned} p_t + vp_x + c^2\rho v_x &= 0 \\ \rho (v_t \,{\color{blue}+}\, v v_x) + p_x &= 0 \end{aligned} We rewrite this as ${\bf p}_t + {\bf A}({\bf p}) {\bf p}_x = {\bf 0}$ for ${\bf p} = (p, v)^\top$, where $$ {\bf A}({\bf p}) = \begin{pmatrix} v & c^2\rho \\ 1/\rho & v \end{pmatrix} \sim \begin{pmatrix} v-c & 0 \\ 0 & v+c \end{pmatrix} $$ has the eigenvectors $(\pm\rho c, 1)^\top$ corresponding to the eigenvalues $\lambda^\pm = v\pm c$. This gives us the Riemann invariants $w^\pm = p \mp \rho c v$, which satisfy the transport equations $$ (w^\pm)_t + \lambda^\mp\, (w^\pm)_x = 0 $$ to be solved by using the method of characteristics.