Generalized Hamiltonian formalism for spatially nonlocal nonlinear differential equations
Abstract
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.
Related
- Nonlinear differential-algebraic systems with dissipative port-Hamiltonian structure
- A Generalized Scalar Auxiliary Variable Method for Structure-Preserving and Efficient Integration of Nonlinear port-Hamiltonian DAEs
- Painlevé Integrability, Hamiltonian Structure and Exact Elliptic Reductions of a Generalized Nonlinear Wave Equation
- Koopman-von Neumann formalism meets contact Hamiltonian systems
- Nonlinear thermodynamic formalism for correspondences
BibTeX
@misc{pazarci2026generalized,
title = {{Generalized Hamiltonian formalism for spatially nonlocal nonlinear differential equations}},
author = {Ali Pazarci and Nadir Ghazanfari and Ilmar Gahramanov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22066},
eprint = {2609.22066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22066},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized Hamiltonian formalism for spatially nonlocal nonlinear differential equations
AU - Ali Pazarci
AU - Nadir Ghazanfari
AU - Ilmar Gahramanov
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22066
UR - https://arxiv.org/abs/2609.22066
AB - 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.
ER -