Approximations for KdV equation as a Hamiltonian system

KdV equation is one example of infinite-dimensional Hamiltonian systems. In this paper, we consider a Fourier spectral approximation and a collocation approximation for the KdV equation with emphasis on its Hamiltonian nature. In particular, we analyze the relations between the Fourier spectral method and the collocation method and in this manner we naturally show that several conservation laws of the KdV equation can be preserved for the collocation method.