Nonlinear Relativistic and Quantum Equations with a Common Type of Solution

TL;DR

This paper introduces generalized nonlinear quantum equations that unify Schrödinger, Klein-Gordon, and Dirac equations, featuring soliton-like solutions expressed via q-exponentials, while maintaining Einstein's energy-momentum relation.

Contribution

It proposes a unified nonlinear framework for key quantum equations with q-dependent terms, revealing a common soliton-like solution structure.

Findings

Equations recover standard forms as q approaches 1.

Solutions are expressed in terms of q-exponentials from nonextensive statistics.

Energy-momentum relation remains valid for all q values.

Abstract

Generalizations of the three main equations of quantum physics, namely, the Schr\"odinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index $q$, are considered in such a way that the standard linear equations are recovered in the limit $q \rightarrow 1$. Interestingly, these equations present a common, soliton-like, travelling solution, which is written in terms of the $q$-exponential function that naturally emerges within nonextensive statistical mechanics. In all cases, the well-known Einstein energy-momentum relation is preserved for arbitrary values of $q$.