# Homework Help: Sketching the solution to the IVP (Characteristic curves)

1. Oct 2, 2012

### Leb

1. The problem statement, all variables and given/known data
Sketch the solution to the IVP
$u_{t}+uu_{x}=0 \\ u(x,0) = e^{-x^{2}}$

2. Relevant equations
Monge's equations $\frac{dt}{d\tau} = 1 \\ \frac{dx}{d\tau}=u \\ \frac{du}{d\tau}=0$

3. The attempt at a solution
I think I do not really need the Monge's equation, but I have no idea, how to sketch the wave profile. Could someone please explain it to me step by step ?
P.S.
I also got stuck on solving the Monge's equations. Particularly in the part where $dx=u(x,t)d\tau$ I tried taking $t_{0} = 0$ which gave me $t=\tau$ and also taking $u(x,t)=u_{0}=F({\sigma})=const.$ which makes it independent of t. Pluging it back in $dx=u_{0}dt --> x = u_{0}t + x_{0}$ where $x_{0} =x'*u_{0}=const.$ Then I blindly assume $x_(0) = \sigma$ which then gives me that the Characteristic Curves should be of the form $x=u_{0}t+\sigma$. Now, since we are given that at t=0 $u(x,0)=e^{-x^{2}}$ and $u(x,0)=u_{0}(\sigma)=e^{-\sigma^{2}}$. So from this, I naively deduce, that each characteristic passes through $(\sigma,0)$ with gradient $e^{\sigma^{2}}$. Can someone point out which assumptions/steps are wrong ? Thanks.