Derivation of Korteweg-de Vries flow equations from nonlinear Schrödinger equation

We perform a multiple scales analysis on the nonlinear Schrödinger (NLS) equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg-de Vries (KdV) flow equations in the bi-Hamiltonian form with the corresponding Hamiltonian functions.