A representation-theoretical approach to higher-dimensional Lie–Hamilton systems: The symplectic Lie algebra sp(4,R)

IF 3.4 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2024-11-19 DOI:10.1016/j.cnsns.2024.108452

Rutwig Campoamor-Stursberg , Oscar Carballal , Francisco J. Herranz

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引用次数: 0

Abstract

A new procedure for the construction of higher-dimensional Lie–Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of intrinsic Lie–Hamilton system is defined, and a sufficiency criterion for this property given. Novel four-dimensional Lie–Hamilton systems arising from the fundamental representation of the symplectic Lie algebra

s p (4, R)

are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra

h_{6}

and the Lorentz Lie algebra

s o (1, 3)

, are also considered in detail. As applications, coupled time-dependent systems which generalize the Bateman oscillator and the one-dimensional Caldirola–Kanai models are constructed, as well as systems depending on a time-dependent electromagnetic field and generalized coupled oscillators. A superposition rule for these systems, exhibiting interesting symmetry properties, is obtained using the coalgebra method.

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高维 Lie-Hamilton 系统的表征理论方法：交错李代数 sp(4,R)

本文提出了一种构建高维 Lie-Hamilton 系统的新方法。该方法基于属于李代数表示理论的技术及其矢量场的实现。定义了本征 Lie-Hamilton 系统的概念，并给出了这一性质的充分性标准。从交错李代数 sp(4,R) 的基本表示中得到了新的四维李-哈密顿系统，并证明了它们是本征的。此外，还详细研究了两个杰出的子代数--双光子李代数 h6 和洛伦兹李代数 so(1,3)。作为应用，构建了广义贝特曼振荡器和一维卡尔迪罗拉-卡奈模型的耦合时变系统，以及取决于时变电磁场和广义耦合振荡器的系统。利用煤代数方法获得了这些系统的叠加规则，表现出有趣的对称特性。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.