New Type of Soliton Equation Described Some Statistical Distributions and Nonlinear Equations Unified Quantum Statistics

Abstract

We proposed a new type of soliton equation, whose solutions may describe some statistical distributions, for example, Cauchy distribution, normal distribution and student distribution, etc. The equation possesses two characters. Further, from an extension of this type of equation we may obtain the exponential distribution, and the Fermi-Dirac distribution in quantum statistics. Moreover, by using the method of the soliton-solution, the nonlinear Klein-Gordon equation and nonlinear Dirac equations may derive Bose-Einstein and Fermi-Dirac distributions, respectively, and both distributions may be unified by the nonlinear equation.