Hamiltonian formalism for nonlinear Schr\"{o}dinger equations

TL;DR

This paper develops a Hamiltonian formalism for second and fourth order nonlinear Schrödinger equations using the Dirac-Bergmann approach, analyzing constraints and their impact on the equations of motion.

Contribution

It applies the Dirac-Bergmann formalism to nonlinear Schrödinger equations, revealing how degeneracy and higher-order dispersion affect constraint dynamics and Hamiltonian construction.

Findings

Second order equations have only primary constraints.

The form of nonlinearity does not affect constraint dynamics.

Higher order dispersion introduces secondary constraints.

Abstract

We study the Hamiltonian formalism for second order and fourth order nonlinear Schr\"{o}dinger equations. In the case of 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 which contribute to the total Hamiltonian along with their Lagrange multipliers. The order of the Lagrangian degeneracy determines the number of the primary constraints. Multipliers are determined by the time consistency of constraints. If a constraint is not a constant of motion, a secondary constraint is introduced to force the consistency. We show that for both second order nonlinear Schr\"{o}dinger equations we only have primary constraints, and the form of nonlinearity does not change the constraint dynamics of the system. However, introducing a higher order dispersion changes the constraint dynamics and secondary constraints are needed to construct a consistent Hamilton equations of motion.