Generalized Potential and Mathematical Principles of Nonlinear Analysis

TL;DR

This paper introduces a mathematical framework using field theory to analyze nonlinear systems, including chaos and dissipative structures, by establishing generalized potential and evolution laws.

Contribution

It develops a new mathematical concept of generalized potential and clarifies the evolution laws of nonlinear and dissipative systems using field theory.

Findings

Established the concept of generalized potential mathematically.

Clarified the spatiotemporal evolution laws of nonlinear systems.

Proposed evaluation methods for conservation and dissipation in physical fields.

Abstract

In the past hundred years, chaos has always been a mystery to human beings, including the butterfly effect discovered in 1963 and the dissipative structure theory which won the chemistry Nobel Prize in 1977. So far, there is no quantitative mathematical-physical method to solve and analyze these problems. In this paper, the idea of using field theory to study nonlinear systems is put forward, and the concept of generalized potential is established mathematically. The physical essence of generalized potential promoting the development of nonlinear field is extended and the spatiotemporal evolution law of generalized potential is clarified. Then the spatiotemporal evolution law of conservative system and pure dissipative system is clarified. Acceleration field, conservative vector field and dissipation vector field are established to evaluate the degree of conservation and dissipation of physical field. Finally, the development route of new field research and the precondition of promoting engineering application in the future are discussed.