Consider the nonlinear wave problem
$$
\begin{cases}
u_t + (x+u)u_x = 0\\
u(x,0) = x
\end{cases}
$$a) Write down Monge's equations in the for $du/d\tau$…,etc, and solve them.
b) Plot the characteristics in the tx plane
So, I honestly don't know how to do this. I know the process when we have an equation of the form $u_t + c(u)u_x = 0$, but not when that function also includes x?
All I can confidently do is write Monge's equations for the problem:
$$
\begin{cases}
\dfrac{dt}{d\tau} = 1,\\
\dfrac{dx}{d\tau} = x+u,\\
\dfrac{du}{d\tau}=0
\end{cases}
$$
Which gives that
$$
\begin{cases}
u = c_1 \text{ (const. w.r.t } \tau)\text{ and }
\\\dfrac{dx}{dt} = x + u
\end{cases}
$$
But I don't know how to go about solving this and I'm struggling to find resources on this.
If anyone could explain what I need to do here or point me to some resources on this it would be much appreciated.
I tried to follow this answer, as it seemed like a similar problem, however the solution I got was
$$
u(x,y) = \frac{x(1-t)}{1+t},
$$ but this doesn't work when substituted into the PDE so I must have done something wrong or perhaps that method is not what I want to be doing?
Best Answer
Regarding the PDE
$$ \begin{cases} u_t + (x+u)u_x = 0\\ u(x,0) = x \end{cases} $$
we have
$$ \frac{dt}{1}=\frac{dx}{x+u}=\frac{du}{0} $$
and solving,
$$ \cases{ u = c_1\\ e^{t + c_2} = x+c_1 \Rightarrow e^t c_3 = x + c_1 } $$
but $c_1 = u$ and $c_3 = G(u)$, so
$$ e^tG(u) = x+u $$
Now $G(\cdot)$ is determined according to the boundary conditions:
$$ e^0G(u(0,x))=x+u(0,x)\Rightarrow G(x) = 2x $$
then we follow with
$$ e^t(2 u)=x+u\Rightarrow u(t,x) = \frac{x}{2e^t-1} $$
Along the characteristic curves, $u(x,t) = c_0$ so we have