Discrete KdV Equation Under Periodic Boundary Condition

Polynomials and the theory of integrable systems have a deep connection.For exam-ple, the connection between KdV equation and diferential polynomial Zn of its conservedquantity, KdV equation and Faulhaber polynomials, KdV equation and Adler-Moser poly-nomials, The AKNS series that Li NianHua and Master Li YuQi found and a diferentialpolynomial series and so on. In this paper, through studying of discrete KdV equationwith periodic boundary Solution, we fnd a new multivariate polynomial series. Thisnew multivariate polynomial series has a wealth of multivariate polynomial mathematicalknot Structure. The general solution of the discrete KdV narrow sense can be determinedby the sequence of compact polynomial represented. This paper is constructed of fourchapters as follow:Chapter1Introduction.About integrable system which has a rich mathematical struc-ture.Specially, The special structure of the polynomial occupies an important position.Furthermore,about some introduction of the topic of this paper and the main work andrelated prior knowledge.Chapter2About describing the periodic boundary discrete KdV equation whichcontains a new multivariate polynomial sequence.Chapter3About describing the discrete KdV equation with periodic boundary nar-rowly general solution and its proof.Chapter4About conclusion and outlook.