Hamiltonian formalism for nonlinear Schrödinger equations

Abstract

We study the Hamiltonian formalism for second and fourth order nonlinear Schrödinger equations. In the case of the second order equation, we consider cubic and logarithmic nonlinearities. Since the Lagrangians generating these nonlinear equations are degenerate, we follow the Dirac–Bergmann formalism to construct their corresponding Hamiltonians. In order to obtain consistent equations of motion, the Dirac–Bergmann formalism imposes some set of constraints that contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of primary constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency condition. We show that for second order and fourth order nonlinear Schrödinger equations we only have primary constraints, and the form of nonlinearity or the order of derivatives does not change the constraint dynamics of the system. However, we observe that introducing new fields to treat higher derivatives in the Lagrangians of these equations changes the constraint dynamics, and secondary constraints are needed to construct a consistent set of Hamilton equations.