The development of the theory might provide an analysis of the mathematical solutions of "nonlinear" physics and, in particular, of "quasi particles ".
In physics the quasi particle is a separate type of particle that can be identified in physical systems containing interacting particles. The quasi particle can be thought of as the set of single particle and the surrounding cloud (hence also the term synonymous with dressed particle) consists of other particles, blown away or dragged by the particle in its motion through the system. Is that the entire amount can be considered as an effective free particle (non-interacting).
The “soliton” is a quasi particle. In mathematics and physics a soliton is a self-reinforcing solitary wave caused by nonlinear effects in a medium of propagation. Solitons are found in many physical phenomena, since they emerge as solutions to a broad class of partial differential equations that describe many nonlinear physical phenomena. The phenomenon of solitons was first described by John Scott Russell that observed a solitary wave upstream in the canal for miles without losing energy, reproduced the phenomenon in a container and so called "Wave of Translation".
It is not easy to define precisely what a soliton is. Drazin and Johnson (1989) describe solitons as solutions of nonlinear differential equations that:
1. describe waves of permanent form;
2. are localized, so that decay or approximate a constant at infinity;
3. can interact strongly with other solitons, but emerge from collisions unchanged except for a phase shift.
"A major problem is the possibile identication between dromions and elementary particles and indeed de Broglie, Bohm and others hoped for the explanation of quantum mechanics through nonlinear classic effects. Notably among others the Skyrme model describes nucleons and nucleon-nucleon interactions, while topological solitons give rise to quantization of charges.
A localized and stable wave might be a good model for elementary particles, but we have seen that in nonlinear field equations there is a great variety of coherent solutions and chaotic and fractal patterns. If particles are excitations of nonlinear fields, it is clear that they are not
the only possible excitations".