Let $\xi=x-ct$. Moreover, let $U(\xi)$ be a travelling wave solution of a PDE. Suppose that $U(\xi,t)$ is a solution of a PDE.
The travelling wave $U(\xi)$ is called stable (with respect to the PDE) if there is a neighborhood $N$ of it, such that for a solution $U(\xi,t)$ whose initial value $U(\xi,0)$ is within $N$, there exists some $k\in\mathbb{R}$ such that $$ \lVert U(\xi,t)-U(\xi+k)\rVert\to 0~\text{ as }t\to+\infty. $$
In other words, a travelling wave $U(\xi)$ is stable, if each solution whose initial values are sufficiently close to it (in some norm), converges to a translate travelling wave $U(\cdot +k)$ as $t\to +\infty$.
I am wondering a bit why this reflects a stable behaviour of $U(\xi)$ since the translate travelling wave $U(\cdot +k)$ can be very far away from $U(\xi)$, or not? Maybe this is not possible because of the wave structure, that is, after some time there is the next "maximum" (when the wave is starting again) and so maybe the definition tells us that, indeed, this means that the solution will be close to $U(\xi)$ as $t\to\infty$ if it is close to some translate of it. Is this the reason why it is a stable behaviour?
In other words: If I think of a typical wave form, then, when a solution starts near a wave and then, as $t\to\infty$, it is near some translate of the wave, this implies that it is somewhere between to "maxima" of the wave, so it is quite near to the original wave again?
But what if two maxima of the wave are far away from each other? Then being close to some translate of the wave does not need to mean being near the original wave? Maybe my intuition is wrong.
I advise you to read this review article:
B Sandstede, Stability of travelling waves, in B. Fiedler (ed), 'Handbook of Dynamical Systems', vol 2 (2002), ch. 18, pp 983-1055, DOI.