空间非局部非线性微分方程的广义哈密顿形式体系
Generalized Hamiltonian formalism for spatially nonlocal nonlinear differential equations
- Bogazici University（博阿齐西大学）
- Istinye University（伊斯蒂耶大学）
- Khazar University（哈扎尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文为空间非局部非线性场论提出广义哈密顿形式体系，推导广义欧拉-拉格朗日方程，并成功应用于三个非局部非线性薛定谔方程，建立统一框架。
AI中文摘要：
本文针对拉格朗日密度显式依赖于局部场及其空间反射对应场的空间非局部场论，发展了一种广义哈密顿形式体系。我们从广义变分原理出发，推导出广义欧拉-拉格朗日方程，并引入一种能一致考虑反射场贡献的广义泛函导数。该形式体系被应用于三个空间非局部非线性薛定谔方程。对于Ablowitz-Musslimani方程，我们构建了据我们所知首个直接用复场表述的标准拉格朗日密度，并推导出其完整的哈密顿公式。随后，同一框架被应用于Velasco-Juan和Fujioka引入的两个拉格朗日非局部非线性薛定谔方程，得到了一致的哈密顿公式，这些公式重现了相应的广义欧拉-拉格朗日方程。这些结果为一大类空间非局部非线性场论建立了一个统一的哈密顿框架。
英文摘要：
In this work, we develop a generalized Hamiltonian formalism for spatially nonlocal field theories whose Lagrangian densities depend explicitly on both the local field and its spatially reflected counterpart. Starting from a generalized variational principle, we derive generalized Euler-Lagrange equations and introduce a generalized functional derivative that consistently accounts for reflected-field contributions. The proposed formalism is applied to three spatially nonlocal nonlinear Schrödinger equations. For the Ablowitz-Musslimani equation, we construct, to the best of our knowledge, the first standard Lagrangian density formulated directly in terms of the complex fields and derive its complete Hamiltonian formulation. The same framework is subsequently applied to the two Lagrangian nonlocal nonlinear Schrödinger equations introduced by Velasco-Juan and Fujioka, yielding consistent Hamiltonian formulations that reproduce the corresponding generalized Euler-Lagrange equations. These results establish a unified Hamiltonian framework for a broad class of spatially nonlocal nonlinear field theories.
