Symplectic Computation of the Nonlinear Schrodinger Equation

摘要

We consider the nonlinear cubic Schrodinger equation (NLSE) with initial condition: {iW_t +W_(xx)+a|W|~2W=0 W(x,0) = W_0(x)where x∈R, a is a constant, W(x, t)) is a complex function and W_0 (± ∞) = 0. We choose a popular spatial discretization:iW_t~((l))+ (W~((l+1))-2W~((l))+W~((l-1)))/h~2+a |W~((l))|~2W~((l)) =0where h is the spatial step size and W~((l))(t) = W(lh,t), l = -n,...,-1,0,1,...,n. The discrete model produces a Hamiltonian system:dz/dt = J~(-1)ΔH(z) where H(z) = H(p,q) = 1/(2h~2)[p~T Bp + q~T Bq]+ a/4 ∑_(k=-n)~n [(p)~((k))~2 +(q~((k)))~2]~2.The Hamiltonian can be splitted into three parts, each of which can be explicitly integrated. We use some explicit symplectic scheme to simulate the soliton motions. The conservativity of the scheme for the invariants of the invariants of NLSE and the formal energy of the scheme itself is also tested.