Given the quasilinear equation $u_{x} + u_{y} = e^{-u}$ with the initial conditions
- $x(0, s) = s$
- $y(0, s) = -s$
- $u(0, s) = \ln(s^{2})$
how can I write the characteristic equation where
- $a(x, y, u) u_{x} + b(x, y, u) u_{y} = c(x, y, u)$
- $\frac{dx}{dt} = a(x, y, u)$
- $\frac{dy}{dt} = b(x, y, u)$
- $\frac{du}{dt} = c(x, y, u)$
I took a look at Yehuda Pinchover & Jacob Rubenstein's An Introduction to Partial Differential Equations; the book that my instructor was using, but it was of no help. I am unsure on what am I suppose to do with $x_{t} = a(x, y, u)$ and $x(0, s) = s$ since $x_{t}$ doesn't have an interval/values that I can solve for.
I'd appreciate any guidance.
In your case $a=1$ $b=1$ and $c(u)=e^{-u}$. Therefore, the characteristic system is $$ dx/dt=1, \quad dy/dt=1,\quad du/dt=e^{-u}, $$ so the (projections of) the characteristics on the $XY$-plane are straight lines. Along those, $u$ evolves according to the last equation.