Integrable Reaction - Diffusion Equation Covariant Extension Structure

Wahlquist and Estabrook proposed a theory of prolongation structures of the nonlinear evolution equations(NEEs) in terms of Cartan's exterior differential form method and successfully applied it to kdv equation.Following them,many workers had investigated many nonlinear differential equations by means of differential geometry and group theory and obtained a lot of important results. Based upon the requirement of covariance of the geometrical quantities,Guo et al proposed the covariant theory for the prolongation structure by means of the connection theory of fiber bundles,thus making the whole theory covariant.In this paper we'll use this covariant theory to discuss the prolongation structure of the integrable the inhomogeneous equation of the reaction-diffusion type: By establishing SL(2,R)principal bundle and its associated bundle and finding both 1-dimensional realization and 2-dimensional realization of its algera ,we can successfully give its AKNS-equations and Backlund transformation.