Constructions Of Several Variable-coefficient Integrable Systems And The Related Problems

Integrable system is one of the main contents of the contemporary nonlinear science, which has achieved widely research and application in various fields such as mathematics, physics, biology and communication. Integrable system has experienced a long historical development process, through the study of integrability equation we can prove whether or not the solving new equations are of solvability and whether or not they have certain physical meaning. In this paper, it is based on some existing equation hierarchies to obtain several new integrable equation hierarchies with variable coefficients, which include some existing constant-coefficient integrable systems as special cases. Using such variable-coefficient equations to simulate the nonlinear physical phenomena will be more close to reality. The main results of this paper are summarized as follows:First of all,it is based on the existing loop algebra1~A, we construct new isospectral problem with arbitrary functions to obtain a new hierarchy of integrable system by using Tu format. We then prove the Liouville integrability of the hierarchy of integrable system. As examples, we improve the process of deriving the KN-like equation hierarchy, the AKNS equation hierarchy and the TC equation hierarchy. As a result, new variable-coefficient KN-like equation hierarchy, new variable-coefficient AKNS equation hierarchy and new variable-coefficient TC equation hierarchy are obtained.Second, using the direct sum operation and the isomorphism relation to extend above derived new variable-coefficient KN-like equation hierarchy, AKNS equation hierarchy, TC hierarchy of equations. And then several kinds of new extended integrable coupling models of the equation hierarchies are obtainedFinally, by introducing some coefficient functions we derive a new system of discrete soliton equations with variable coefficients from the discrete formal AKNS spectral problem. As special cases, this paper gives several reduced forms of the derived system of discrete soliton equations, including the famous Toda lattice equation. At the same time, a particular solution of one reduced system of equations has been obtained in the process of reduction.