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. The result is a solution that looks *approximately* like

$$ u(t,x) \approx \begin{cases}
a_1 & x \in (-\infty,c_1t - R) \\
\phi_1(x-c_1t) & x \in (c_1t - R,c_1t + R) \\
a_2 & x \in (c_1t+R,c_2t - R) \\
\phi_2(x-c_2t) & x \in (c_2t - R,c_2t + R) \\
a_3 & x \in (c_2t + R,+\infty) \\
\end{cases} $$

for some appropriately large $R>0$, and $\phi_1$ and $\phi_2$ are the traveling waves profiles, with $\phi_1(-\infty)=a_1$, $\phi_1(+\infty)=\phi_2(-\infty)=a_2$, and $\phi_2(+\infty)=a_3$.

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 $t \to \infty$.