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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Bibliographic Details
Published in:Journal of nonlinear science Vol. 35; no. 6; p. 115
Main Author: Cheng, Pengcheng
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2025
Springer Nature B.V
Subjects:
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 Access:Get full text
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Summary: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.
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ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-025-10215-x