Symplectic Discretization of Hamiltonian Integral Equations with Applications to Nonlinear Wave Dynamics and Relativistic Orbits

We introduce a novel class of symplectic numerical schemes for solving nonlinear Volterra-type integral equations that arise from Hamiltonian systems subject to non-local interactions. The continuous integral equation is characterized by a Hamiltonian function H ( y ) and a kernel K ( t , s , y )...

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Veröffentlicht in:	Journal of nonlinear science Jg. 35; H. 6; S. 115
1. Verfasser:	Cheng, Pengcheng
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.12.2025 Springer Nature B.V
Schlagworte:	Accuracy Algorithms Analysis Classical Mechanics Collocation methods Economic Theory/Quantitative Economics/Mathematical Methods Energy conservation Error analysis Hamiltonian functions Hydrogen sulfide Integral equations Iterative methods Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Nonlinear dynamics Numerical analysis Orbital mechanics Ordinary differential equations Physics Relativistic effects Schrodinger equation Simulation Solitary waves Symmetry Theoretical Gauss–Legendre collocation 65P10 Structure-preserving algorithms 65R20 Hamiltonian integral equations Symplectic discretization Volterra integral equations Geometric numerical integration
ISSN:	0938-8974, 1432-1467
Online-Zugang:	Volltext
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Zusammenfassung:	We introduce a novel class of symplectic numerical schemes for solving nonlinear Volterra-type integral equations that arise from Hamiltonian systems subject to non-local interactions. The continuous integral equation is characterized by a Hamiltonian function H ( y ) and a kernel K ( t , s , y ) that satisfies a specific symplectic symmetry condition, K ( t , s , y ) = J ∂ S ( t , s , y ) ∂ y , ensuring the preservation of the phase space symplectic structure. Building upon this continuous framework, we develop high-order implicit symplectic integrators by discretizing the integral equation using an s -stage Gauss–Legendre collocation method. A rigorous proof demonstrates that the proposed numerical schemes are symplectic, i.e., they preserve the discrete symplectic form d y n + 1 ∧ J d y n + 1 = d y n ∧ J d y n . Furthermore, we establish the superconvergence property of these schemes, showing a global error of O ( h 2 s ) . For practical implementation, the resulting nonlinear algebraic equations at each time step are efficiently solved using a Newton–Krylov iterative method, and an adaptive time-stepping strategy based on embedded symplectic pairs is employed to enhance computational efficiency while maintaining numerical accuracy and symplectic fidelity. The superior long-term performance and structural preservation capabilities of the proposed methods are demonstrated through numerical experiments on challenging problems, including the soliton dynamics of the nonlinear Schrödinger equation and a model of relativistic orbital mechanics inspired by the Einstein–Infeld–Hoffmann equations. Comparisons with standard non-symplectic methods highlight the significant advantages of our symplectic approach in terms of energy conservation and phase space trajectory accuracy over extended simulation times.
Bibliographie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0938-8974 1432-1467
DOI:	10.1007/s00332-025-10215-x