Why are soliton solutions interesting mathematically or physically? 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?
 A: It's tough to pin down one specific reason why solitons might be interesting—that will vary based on the interest of the mathematician or scientist curious about them—but I'll list out two which I subjectively consider to be of note.

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*A feature represented by a soliton tends to be self-stabilizing—this is usually as a result of some deep topological property in the underlying nonlinear dynamical system that governs it—and so it is frequently resistant to perturbations. (See, for example, this paper by Bona et al. on the stability of solitons in generalized nonlinear systems that include the classic Korteweg-De Vries system.) This is very important for people who study and apply transport theory; if the feature is some kind of signal, for example, then I know the signal will not become corrupted as it moves through a medium.

*In a very heuristic and imprecise way, soliton solutions suggest that the evolution operator for a system that exhibits solitons behaves like the d'Alembert (wave) operator in some particular region of the solution space. Because of this, a mathematician or scientist could potentially model the behavior of that system as a linear d'Alembertian system (which math has many tools for) plus whatever corrections need to be implemented for soliton interaction + other nonlinearities of interest.

The combination of both of these elements—self-stabilizing, distinctly nonlinear solutions that can nevertheless often be studied using tools from linear PDEs/operators—makes it very attractive to those who want to study nonlinear partial differential equations, which is a very hairy subject to study in general.
A: One item of great interest is the fact that soliton solutions are sometimes exact solutions to nonlinear PDEs. For example, the Inverse Scattering Transform (IST) method maps the initial nonlinear PDE to a system of linear ODEs via the forward scattering direction, followed by the time evolution solution of that system, and ending with the inverse scattering direction (perhaps the Gel'fand-Levitan-Marchenko integral equation). You can use IST on a number of soliton-admitting PDEs, including KdV, nonlinear Schrodinger, and a few others.
Being able to solve any nonlinear PDE exactly is nearly miraculous. Imagine if someone were able to use the Inverse Scattering Transform method to solve Navier-Stokes? I'm fairly certain that is an active area of research, and it would certainly have immense practical applications.
aghostinthefigures has pointed out the self-correcting nature of soliton solutions. I would add this detail: that is the reason fiber optics are the backbone of the Internet - solitons in fiber optic cables. You don't need repeaters nearly as often if you can transmit data with solitons.
