Non-Linear Schrödinger Equation, Burger’s Equation and Superposition of Solutions

Abstract

We observe that the classical Hamilton-Jacobi and Continuity equations imply that a classical description of a particle-field entity is equivalent to a Non-Linear Quantum Mechanics described by a Schrödinger-type non-linear equation. This would be different from Bohm’s theory. in which one generalizes the Classical Mechanical Hamilton-Jacobi equation to obtain the standard Linear Quantum Mechanical Schrödinger equation.

We studied the symmetry groups of these equations and obtained a differentiable superposition rule for their complete solutions only. Deformation of one Schrödinger equation into another by variation of stochastic parameter γ will suffer a discontinuity at γ = 1 in the sense that the dimensionality of the symmetry group changes from infinity to finite. In our approach. the phase equation leads to an inviscid Burger’s equation whose solutions control the amplitude equation. The appearance of Shock for almost all initial conditions implies that for systems of particles an interaction must be switched on in order to avoid multiple-valued solutions.

Causality and Locality in. Modern Physics and Astronomy: Open Questions and Possible Solutions A Symposium to Honour Jean-Pierre Vigier August 25–29, 1997, York University, Canada