I've seen a great deal about classes of PDEs, particularly dispersive equations (e.g. KdV, NLS, Sine-Gordon) which permit soliton, or wave-packet solutions. My understanding of why these might be interesting is that they behave in a particle-like manner, experiencing things like elastic collisions and the like.
Furthermore in cases like NLS, there is the soliton resolution conjecture which seems to roughly claim that the nonlinearity decomposes into a linear evolution plus soliton solutions.
I have a rather vague question around this, being rather out of the loop, which is something like: why is the existence of soliton solutions mathematically or physically interesting in a broad sense? How come equations with these types of solutions are very popular to study these days?
Secretly, I'm wondering: do soliton solutions help us characterize nonlinear equations in some broad sense the same way spectral theory does for linear equations?