Solving nonlinear inhomogeneous first-order PDE using method of characteristics I am trying to solve the partial differential equation
$$\frac{\partial \rho}{\partial t}+(A+2B\rho)\frac{\partial \rho }{\partial x}=0$$
where $A$ and $B$ are constants. This is meant to model the density of traffic $x$ metres behind some traffic lights $t$ seconds after the light turns green.
My characteristics are the following:
$$\frac{\mathrm{d}t}{\mathrm{d}s}=1,\qquad \frac{\mathrm{d}x}{\mathrm{d}s}=A+2B\rho,\qquad\frac{\mathrm{d}\rho}{\mathrm{d}s}=0.$$
This means that $t=t_0+s$ and that $\rho = \rho_0$. Since $t_0=0$ (the initial time), $t = s$.
I want to impose that the density is a constant (say $\rho_{\max}$) at $t=0$, as in, the cars are all as close to each other as possible behind the traffic light before they start moving. However, to me this suggests that $\rho_0 = \rho_{\max}$, which suggests that $\rho\equiv\rho_{\max}$. This doesn't make sense to me in the context of the problem.
Is this a problem with using the method, or have I simply not set up my initial condition correctly? 
 A: Making $u = A+2B \rho$ we have an equivalent PDE
$$
u_t + u u_x = 0
$$
now making $u(x,t) = X(x)T(t)$ we have
$$
X(x)T'(t)+X(x)T(t) X'(x)T(t)=0
$$
or
$$
\frac{T'(t)}{T^2(t)}+X'(x) = 0\Rightarrow \frac{T'(t)}{T^2(t)} = k = -X'(x)
$$
etc.
A: You did pose the system well, but maybe it's easier to get the general solution by means of the following auxiliar system of ODEs:
$\dfrac{dt}{1}=\dfrac{dx}{A+2B\rho}=\dfrac{d\rho}{0}$
The last ratio means that $\rho=c_1$ for some constant $c_1$. We can use this fact and the first proportion to solve it.
$\dfrac{dt}{1}=\dfrac{dx}{A+2Bc_1}$
$(A+2B\rho)t=x+c_2$
At last, a relation between $c_1$ and $c_2$ must exist: $c_2=f(c_1)$ (differentiable, single variable function $f$), leading to the general solution:
$(A+2B\rho)t=x+f(\rho)$
Your questions about change in density only can be answered when the initial conditions are known. E.g. a constant density for some range for $x$, at $t=0$, and $0$ elsewhere. That kind of problems need some amount of extra assumptions to be solved.
