New Periodic Travelling Waves In Infinite Lattice

This article is concerned with travelling waves in infinite lattice systems.We consider a one dimension chain of particles subjected to a potential f and nearest neighbor (non-linear) interaction potential V. The dynamics of the system is described by the infinite system of second order ordinary differential equations: q(n,t)+f’(q(n,t))=V’(q(n+l,t)-q(n,t))-V’(q(n,t)-q(n-l,t)), t E R, n E Z (4) Where f,V∈C1(R).A travelling wave,say with speed c>0and profile u,is a solution of (4) of the form q(n,t)=u(n－ct), nEZ (5) Plugging the Ansatz (5) into (4) yields the second order backward-forward differential equation for the wave profile: c2u"+f’(u)=V’(Au)-V’(Au)(6) where the difference operators A andA are defined by Au(s)=u(s+1)-u(s)=A(s+1) s∈R. As is well known in the Calculus of Variations,solutions of (6) can be obtained as critical points of the action functional (?) defined on some appropriate Hilbert space,where k (?) R. When the Palais-Smale com-pactness condition is satisfied,the critical point of J can be detected with the aid of the mountain pass theorem,or some of its variants.First of all in the second chapter, under the appropriate guard potential function condition; equation(6) exists a negative travelling wave by Mountain pass theorem,and make use of the Linking theorem get a non-constant travelling wave solutions of equation (6). Second in the third chapter, optimize our solutions, we get the existence of monotone travelling wave by some explicit methods. Then in the chapter4of the grid system is proof the ground the existence and convergence of the ground wave.