The Conserved Density Of A Class Of Differential-diffrence Equations

In recent years because of natural the subjects such as sociology, medicine, biology, finance and the development of the edge discipline, many problems often boils down to the differential difference equations mathematical problems. A number of physically interesting problems can be modeled with nonlinear DDEs, for example, particle vibrations in lattices, currents in electrical networks,pulses in biological chains, etc. In addition, with the develop-ment of the electronic computer, some differential equations can be discretized, thus forming differential difference equations, and then with the aid of computer calculation and numerical analysis, therefore differential difference equation in the numerical simulation of nonlinear partial differential equation, queuing problem and discretization of solid state, and also plays an important role in quantum physics. Research on difference differential equation can be traced back to the 1950s at the earliest, Fermi, Pasta, and Ulam study the famous Fermi-Pasta-Ulam problem. Since then, the differential difference equation becomes the focus of many nonlinear studys and received wide attention of scholars both at home and abroad.Differential difference equation with discrete difference equation is different, it is a dis-crete, or some or all of the space variable is discrete, and usually time variable is continuous, so the research method and the traditional classical differential equation and differential e-quation is a very different nature. Yamilov,Cherdantsev,Shabat and a group of scholars make the deep research of differential difference equations, mainly including investigations of inte-grability criteria, the computation of densities, generalized and formal symmetries, recursion operators, etc.This paper mainly uses the generalized symmetries method and the direct method to research the conserved density of the Volterra type difference differential equations. This paper has the following structure. The first chapter introduces the research background of the difference differential equations and the main results of this article; The second chapter mainly introduces two methods of research of differential difference equations:generalized symmetric method, form symmetric method, as well as the relevant results; In the third chap-ter, we mainly study the conserved density of a Volterra type differential diffrence equations as a system, the main conclusions are as follows.In this paper, we consider the Volterra-type difference differential equations below where p(un) is a polynomial about un. Using the shift operators D, (1) can be simplified asFirst of all, we consider the conserved density of (2) in the form of p= p(u) and obtain the following theorem.定理0.1 is the conserved density of (2).The second, we consider the conserved density of (2) in the form of p= p(u, Du).定理0.2 There is the conserved density of (2) in the form of p= p(u, Du) if and only if deg p(u)< 2, where deg p(u) is the polynomial order of p(u).According to theorem (0.2), we knows that there is no conserved density of (2) in the form of p= p(u, Du) when deg p(u)> 2.Further, we have the following conclusion.定理0.3 Assuming p(u)= au2+bu+c, a≠0. If there is the conserved density of (2) in the form of p= p(u, Du), then and the corresponding flux where α,β,γ,δ is any constant,when deg p(u)=1, the conserved density and the corresponding flux of (2) is obtained in the [2,4].The last, using the method of Hickman [29], we consider the conserved density of (2) in the form of ρ=ρ(Dpu, Dp+1u,…, Dqu).定理0.4 If ρ=ρ(Dpu,Dp+1u,…,Dqu)is the conserved density of (2),then p satisfys where p,q is a natural number,p<q,q>1.