The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.
The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.