This article proposes a set of ideas concerning the introduction
of nonlinear analysis, particularly nonlinear PDE, in the theory of particles.
The Quantum Mechanics theory is essentially a linear theory, due mainly to
the fact that there was a the lack of nonlinear mathematics at the time of the
discoveries in particle physics. The main idea is to perturb the Schroedinger
equation by a nonlinear term. This nonlinear term has two main parts, a
second order quasilinear differential operator responsible for the smoothing
of the solutions and a nonlinear 0-order term with a singularity providing
topology to the space. By minimizing the energy functional, solutions to the
equation are obtained in each topological class. Then the qualitative properties
of the soliton is analyzed. By rescaling arguments, the asymptotic behavior of
the static solutions is studied. Next the evolution is studied, deriving stability
of the soliton and the guidance formula. In this way the equations of Bohmian
Mechanics are obtained. Most proofs are omitted, but in all cases a proper
reference is given.