I would like to solve the 1D nonlinear advection equation with the method of characteristics. Here is my notation:
\begin{equation} \begin{cases} \rho_t + (1+\rho)\rho_x = 0\\ \rho = \rho(x,t); \quad \rho(x,0) = \frac{1}{1+x^2} \end{cases} \end{equation} What I have been up to is the following, using the parameter s: \begin{align} &\frac{d\rho}{dt} = \frac{dx}{ds}\frac{\partial \rho}{\partial x} + \frac{dt}{ds}\frac{\partial \rho}{\partial t} = 0\\ \Longrightarrow \; & \frac{dt}{ds} = 1 ;\quad t(0)=0 \; \Longrightarrow \; t=s \\ \Longrightarrow \; & \frac{d\rho}{ds} = 0 ;\quad \rho(0)=\rho_0 \; \Longrightarrow \; \rho = \rho_0 \\ \Longrightarrow \; & \frac{dx}{ds}=1+\rho= 1+\rho_0 ;\quad x(0) = f(\rho_0) \; \Longrightarrow \; x = (1+\rho_0)s + f(\rho_0) \end{align} So that I end up with \begin{equation} f(\rho) = x - (1+\rho)t \; \Longrightarrow \; \rho = F(x - [1+\rho]t ) \end{equation} I tried to apply several techniques I found to find the solution of the above Riemann problem using the method of characteristics. I have the general form of the solution as F but I would like to have the analytic solution for this case so I can plot for several times and see the shockwave.
Note: This problem comes from this online document, chapter 11.
Your calculus is correct. You found the general solution : $$\rho=F\left(x-(1+\rho)t\right)$$ Condition : $$\rho(x,0)=\frac{1}{1+x^2}=F\left(x-(1+\rho)0\right)=F\left(x\right)$$ The function $F$ is determined : $$F(X)=\frac{1}{1+X^2}$$ We put this function into the general solution where $X=x-(1+\rho)t$ $$\rho=\frac{1}{1+(x-(1+\rho)t)^2}$$ This is the solution on implicit form.
In order to obtain the explicit solution, solve the cubic equation for $\rho$ : $$(1+(x-(1+\rho)t)^2)\rho-1=0$$
ADITION after the discussion in comments.
There is no difficulty to plot the figures that you saw in the document : https://courses.physics.ucsd.edu/2011/Spring/physics221a/LECTURES/CH11_SHOCKS.pdf
For exemple, to plot the curves corresponding tu figure 11.1 : $$x=(1+\rho)t\pm\sqrt{\frac{1}{\rho}-1}$$ Plot the two branches with signs $+$ and $-$.
Inverse the axis if you want it in the standard position.