标题： 非线性薛定谔方程的哈密顿形式 显示英文标题

标题： Hamiltonian formalism for nonlinear Schrödinger equations

摘要： 我们研究二阶和四阶非线性薛定谔方程的哈密顿形式。 在二阶方程的情况下，我们考虑立方和对数非线性项。 由于生成这些非线性方程的拉格朗日量是退化的，我们遵循狄拉克-伯格曼形式来构造它们对应的哈密顿量。 为了获得一致的运动方程，狄拉克-伯格曼形式施加了一些约束集，这些约束与它们的拉格朗日乘子一起贡献到总哈密顿量中。 拉格朗日量退化程度决定了主约束的数量。 乘子由约束的时间一致性确定。 如果一个约束不是运动常数，则引入次级约束以强制一致性。 我们证明对于两种二阶非线性薛定谔方程，我们只有主约束，而非线性项的形式不会改变系统的约束动力学。 然而，引入更高阶的色散会改变约束动力学，需要次级约束来构建一致的哈密顿运动方程。 显示英文摘要

摘要： 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.