Traveling wave

Remarks

In $\R^n$ one also considers plane traveling waves $u(x, t) = U(x \cdot e - ct)$ for a unit vector $e$. Substituting $\xi = x - ct$ turns the equation into an ordinary differential equation for $U(\xi)$, in which $c$ is an unknown parameter.

Example: the viscous Burgers' equation $u_t + u u_x = \nu u_{xx}$, with $\nu > 0$, has the front $u = c\left(1 - \tanh\frac{c(x - ct)}{2\nu}\right)$ for every $c > 0$. Check: with $U(\xi) = c(1 - \tanh \tfrac{c\xi}{2\nu})$, the equation becomes $-cU' + UU' = \nu U''$. Integrating once with $U(+\infty) = 0$ gives $\nu U' = \tfrac12 U^2 - cU$, which this $U$ satisfies. The front goes from $2c$ to $0$, and its speed $c$ is the average of the two states, as for a shock.

Non-example: $u(x, t) = e^{-t} \sin x$ solves the heat equation $u_t = u_{xx}$, but it is not a traveling wave. For a traveling wave, $\sup_x \abs{u(x, t)} = \sup \abs{U}$ does not depend on $t$, while here it equals $e^{-t}$.