Confusion about notation for reflected travelling wave solutions I have a rather stupid question, I guess.
For some PDE, it is said that there exists a right-moving travelling wave solution $u(x,t)=\phi(\xi)$ with $\xi=\pm x-ct, c>0$ and $\lim_{\xi\to-\infty}\phi(\xi)=2\pi, \lim_{\xi\to\infty}\phi(\xi)=0$. It is also said that, upon reflection, there is a left-moving travelling wave solution.
I am a bit confused about two things.
1.) What about the $\pm$-sign in $\xi=\pm x-ct, c>0$? Of course, if I define $\xi=x-ct$ for $c>0$, then $\phi(\xi)$ is right-moving and if I assume that for this $\xi$, $\lim_{\xi\to-\infty}\phi(\xi)=2\pi$ and $\lim_{\xi\to\infty}\phi(\xi)=0$, then isn't $\phi(-x-ct)$ still right-moving with the only difference that it is now an increasing solution?
2.) What is meant with the reflection? For a left-moving solution $\psi$, I need $\psi(\eta)$ with $\eta=x+ct$ and $c>0$. In which sense is this a reflection of $\phi(\pm x-ct)$?
 A: *

*You can think about the direction of propagation as follows. Consider that the solution takes the form $u(x,t) = \phi(\pm x - c t)$ at some time $t$. Now, we increase time from $t$ to $t + \Delta t$ with $\Delta t>0$, which gives us the waveform
\begin{aligned}
u(x, t+\Delta t) &= \phi(\pm x - c (t + \Delta t)) \\
&= \phi(\pm (x \mp c \Delta t) - c t) \\
&= u(x \mp c \Delta t, t) .
\end{aligned}
In other words, the value of $u$ at $(x, t+\Delta t)$ is the same as that at abscissa $x \mp c \Delta t$ and earlier time $t$. Given that $c\Delta t>0$, this shows that information is traveling towards increasing $x$ with increasing times if the upper sign is used $(\xi = x-ct)$, and that information is traveling towards decreasing $x$ otherwise $(\xi = -x-ct)$ -- see also this article for complements. Note that the left-going wave may not simply be a reflection of the right-going wave. In fact, if $u(x,t) = \phi(x-ct)$ solves the PDE problem
$$
F(u, u_x, u_t, u_{xx}, \dots) = 0
$$
with suitable boundary conditions at infinity, then its reflection $u(y,t)$ where $y=-x$ solves
$$
F(u, -u_y, u_t, u_{yy}, \dots) = 0
$$
with reflected boundaries.

*Setting $\psi(\eta) = \phi(-\eta)$ gives you $\psi(x+ct) = \phi(-x-ct)$.

