The Prolongation Structures for the System of the Reaction-Diffusion Type

Xiaojuan Duan

doi:10.4236/jamp.2017.51010

Abstract

Equations of the reaction-diffusion type are very well known and have been extensively studied in many research areas. In this paper, the prolongation structures for the system of the reaction-diffusion type are investigated from theory of coverings. The realizations and the classifications of the one-dimensional coverings of the system are researched. And the corresponding conservation law of the one-dimensional Abelian coverings is concluded, which is closely connected with the symmetry of the system by Noether theorem.

Highlights

Equations of the reaction-diffusion type have been widely studied in many research areas [1] [2]
The corresponding conservation law of the one-dimensional Abelian coverings is concluded, which is closely connected with the symmetry of the system by Noether theorem
Alfinito et al [1] used Wahlquist-Estabrook (WE) prolongation structure theory proposed by Wahlquist and Estabrook [6] [7] and carried out the detailed integrable analysis

Summary

Introduction

Equations of the reaction-diffusion type have been widely studied in many research areas [1] [2]. The approach is called the theory of coverings, which treats a PDE as an (infinite-dimensional) submanifold in the space J ∞ (π ) of infinite jets for a bundle π : E → M whose sections play the role of unknown functions (fields). This attitude allows applying to PDEs powerful techniques of differential geometry and homological algebra. It is noticed that the WE prolongation structures are an essentially special type of coverings [13] [15] Cheng and He successfully gave the realizations and classifications of one-dimensional coverings of the MB (modified Boussinesq) system by using the theory.

Basic Definitions and Statements

The Prolongation Structures of the Reaction-Diffusion System

Concluding Remarks