I am studying conservation laws and reviewing the papers I get a doubt. Consider
$$u_t+f(u)_x=0$$ with $f$ smooth a conservation law and take the characteristics
$$x(t)\,\, ; \,\, x'(t)=f'(u(x(t),t))\,\, ; x(0)=x_0$$
Are they always straight lines for any $f$?
I did the following calculus:
$$x''(t)=f''(u(x(t),t)\underbrace{(u_x x'(t)+u_t)}_{0}=0$$
I think I am wrong, because I've thinked that this assertion that characteristics are lines holds only for some cases, as Burgers' equations.
Many thanks.
Best Answer
Note that $u$ is constant along the characteristic curves $x'(t) = f'(u(x(t),t))$ parameterized by $t$. Indeed, according to the chain rule and the quasi-linear PDE $u_t + f'(u) u_x = 0$ itself, we have $$ \frac{\text d}{\text d t}u(x(t),t) = u_x x'(t) + u_t = 0\, . $$ Thus, those curves are straight lines in the $x$-$t$ plane.