Long Time Stability Of DNLS Equation

The mathematical model of derivative nonlinear Schrodinger equation is very important in physics. In this paper, we study the behavior of the solution of derivative nonlinear Schrodinger equation under the periodic boundary condition u(x+2π, t)=u(x, t). and the potential V belong to the with fixed potential under the periodic boundary condition around the elliptic equilibrium point and obtain that if the initial valve u(x,0) of solution is closed to the equilibrium point, then the solution u(x,t) is also close to the equilibrium point in a large time period under a high index s norm as ||u(x,t)-u(x,0)||s≤2ε.In order to prove the conclusion, first, through Fourier transform put the equa-tion(0.2) into discrete the infinite dimensional equation. The original equation has a certain special form and symmetrical properties, then the infinite dimensional e-quation is the infinite dimensional Hamiltonian system. With the original equation of nonlinear term containing derivative, then the corresponding infinite dimension-al Hamiltonian equations with unbounded disturbance. Using the Birkhoff normal form theory on the infinite dimensional unbounded Hamiltonian system has four times Normal form with the Hamiltonian. The theory need the frequency of the Hamiltonian have a good condition of non resonant. In this paper, the potential V ∈(?) and the 4-times of the non resonant condition. Final.4-times of Birkhoff normal form with symmetry of Hamiltonian can get long time stability on the infi-nite dimensional unbounded Hamiltonian system, and then get long time stability on the(0.2).