# Are characteristics of $u_t+f(u)_x=0$ always straight lines?

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.

## 1 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.

• That's incredible... many thanks !! – Quiet_waters May 14 '19 at 20:49