I am looking for general reference on "stacked traveling waves" or "wave trains", or perhaps wave superpositions. They are a bit like multi-soliton solutions to the KdV equation, except each wave may have different left/right limits at infinity.

For example, the reaction diffusion equation,

$$ \partial_t u = \Delta u + f(u) $$

with $f(u) = u(u-a)(u-1)$. If you take the heaviside initial data, the solution separates into multiple traveling waves. One way to say this is that the limiting function $ x \mapsto \lim_{t\to\infty} u(t,x+tc) $, for various $c \in \mathbb{R}$, is a constant function for all but two values of $c$, which are the velocities of the two waves.

Note that I am \*not\* referring to the bistable equation, in which my $f(u)$ is replaced by $-f(u)$, and has been studied a lot. In *that* equation, there are distinct traveling waves, but they merge into a single traveling wave resulting in the so-called "entire solution": https://link.springer.com/article/10.1007/s10884-006-9046-x. Rather, I am looking to define asymptotic behavior of stacked traveling waves that separate as $t \to \infty$.