Hamiltonian formalism for nonlinear Schrödinger equations

We study the Hamiltonian formalism for second order and fourth nonlinear Schr\"{o}dinger equations. In case of equation, we consider cubic logarithmic nonlinearities. Since Lagrangians generating these equations are degenerate, follow Dirac-Bergmann to construct their corresponding Hamiltonians. obtain consistent motion, imposes some set constraints which contribute total along with Lagrange multipliers. The Lagrangian degeneracy determines number primary constraints. Multipliers determined by time consistency If a constraint is not constant secondary introduced force consistency. show that both only have constraints, form nonlinearity does change dynamics system. However, introducing higher dispersion changes needed Hamilton motion.