Solitons and Bohmian mechanics

Abstract

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.