Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

We derive a Hamiltonian form of the fourth-order (extended) nonlinear Schrödinger equation (NLSE) in Klein–Gordon model with quadratic and cubic nonlinearities. This describes propagation envelope slowly modulated wave packets approximated by superposition fundamental, second, zeroth harmonics. Although extended NLSEs are not generally PDEs, derived here is PDE that preserves structure original equation. could be achieved expressing fundamental harmonic its first derivative symplectic form, second harmonics calculated from variational principle. demonstrate non-Hamiltonian NLSE under discussion can retrieved simple transformation variables.