On A Class Of Toda Type Equations

Toda system is a classic integrable system.In this paper,we study a class of extended Toda equations,and these equations are deeply related to the entire solutions of the Allen-Cahn equation and the Schrodinger equation.Firstly,we review the anti-scattering transformation and related integrable system theory of the classical Toda system,including the backscattering theory of the one-dimensional Toda system,the relationship between the Toda system and the Kd V equation,the Backlund variation and the Lump solution of the two-dimensional Toda system.Secondly,we review the asymptotic behavior of the classical one-dimensional Toda system at infinity.Finally,we study the asymptotic behavior of solutions for a class of two-dimensional Toda systems.Using the theory of ordinary differential equations,we prove that the solutions of these equations are asymptotically linear at infinity.The results of the paper can be used to construct the global solutions of the Allen-Cahn equation and the Schrodinger equation.