One approach to this problem might be via "dark solitons": https://en.wikipedia.org/wiki/Vector_soliton#Vector_dark_soliton "Dark solitons are characterized by being formed from a localized ***reduction of intensity*** compared to a more intense continuous wave background. Scalar dark solitons (linearly polarized dark solitons) can be formed in all normal dispersion fiber lasers mode-locked by the nonlinear polarization rotation method and can be rather stable. *Vector dark solitons are much less stable* due to the cross-interaction between the two polarization components. Therefore, it is interesting to investigate how the polarization state of these two polarization components evolves." "In 2009, the first dark soliton fiber laser has been successfully achieved in an all-normal dispersion erbium-doped ber laser with a polarizer in cavity. Experimentally finding that apart from the bright pulse emission, under appropriate conditions the ber laser could also emit single or multiple dark pulses. Based on numerical simulations we interpret the dark pulse formation in the laser as a result of dark soliton shaping." At 03:05 PM 9/25/2017, Henry Baker wrote:

Solitons use a combination of nonlinearity and dispersion to propagate pulses along a 1-D channel whose *shape is preserved*.

Standard solitons "simply" pass through one another "unaltered in shape, amplitude, or velocity" (see link below).

Are there weird equations and soliton solutions that attempt to preserve the "holes" between them -- i.e., in addition to preserving the shape of the pulse itself, the medium would also attempt to preserve the *space between pulses*, so that they would settle down into an orderly pulse train?

---- https://arxiv.org/abs/1407.5087

Collisions of matter-wave solitons

"Solitons are localised wave disturbances that propagate without changing shape, a result of a nonlinear interaction which compensates for wave packet dispersion. Individual solitons may collide, but a defining feature is that they pass through one another and emerge from the collision unaltered in shape, amplitude, or velocity. This remarkable property is mathematically a consequence of the underlying integrability of the one-dimensional (1D) equations, such as the nonlinear Schr\"odinger equation, that describe solitons in a variety of wave contexts, including matter-waves$^{1,2}$. Here we explore the nature of soliton collisions using Bose-Einstein condensates of atoms with attractive interactions confined to a quasi-one-dimensional waveguide. ..."