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:
I think I am wrong, because I've thinked that this assertion that characteristics are lines holds only for some cases, as Burgers' equations.