Numerical Modeling of Nonlinear Wave Transport in Branched Quantum Graphs with Local Inhomogeneity

A. R. Preobrazhenskaya

Abstract


This paper presents numerical modeling of signal transport described by wave packets of the nonlinear Schrödinger equation through a branched quantum graph with local inhomogeneity on one of the central edges. An explicit finite-difference scheme is employed to obtain the numerical solution, with Dirichlet boundary conditions and Kirchhoff conditions at branching vertices (signal convergence and divergence nodes). The local inhomogeneity is defined by the medium contrast parameter χ, characterizing the degree of environmental heterogeneity and varied in the range from 1 to 8. The dependence of the signal transmission coefficient on χ is investigated.

It is established that even in a homogeneous medium (χ = 1), a topological energy reflection effect occurs: due to the complex branched topology of the graph, approximately 50% of the wave energy is reflected at the branching nodes. With increasing χ, the signal transmission coefficient monotonically decreases. A barrier effect emerges at χ > 6: the local inhomogeneity region becomes a barrier, transmitting only about 10% of the initial energy. Analysis of energy distribution across graph zones at the final time demonstrates interference-enhanced wave function density in the region before the barrier with increasing contrast parameter. The correctness of the observed physical effects is confirmed by comparison with the linear case and convergence tests on the time step. Numerical stability of the scheme is substantiated for the selected modeling parameters. Limitations of the proposed solution method are discussed, and recommendations for further development of the research approach are provided. The obtained results can be applied to control the bandwidth of branched waveguides by varying inhomogeneity parameters.

Full Text:

PDF (Russian)

References


Pokorny Yu.V., Penkin O.M., Pryadiev V.L., Borovskikh A.V., Lazarev K.P., Shabrov S.A. Differential Equations on Geometric Graphs. Moscow: Fizmatlit, 2004, 272 p.

Kuchment P. Quantum graphs: an introduction and a brief survey, 2008, arXiv:0802.3442v1. Available: https://arxiv.org/abs/0802.3442

Do N. T., Kuchment P., Ong B. On resonant spectral gaps in quantum graphs, 2016, arXiv:1601.04774v2. Available: https://arxiv.org/pdf/1601.04774

Berkolaiko G., Kuchment P. Introduction to Quantum Graphs. AMS, Providence, R.I., 2013.

Berkolaiko G. An elementary introduction to quantum graphs, 2016, arXiv:1603.07356v2. Available: https://arxiv.org/pdf/1603.07356

Harrison J.M. Quantum graphs with spin Hamiltonians, 2008, arXiv:0712.0869v2. Available: https://arxiv.org/pdf/0712.0869

Kurasov P. Understanding Quantum Graphs. Acta Physica Polonica A, Institute of Physics, Polish Academy of Sciences, 2019, pp. 797-802.

Exner P., Jex M. On the ground state of quantum graphs with attractive δ-coupling, 2011, arXiv:1110.1800v2. Available: https://arxiv.org/pdf/1110.1800

Exner P., Turek O. Quantum graphs with the Bethe-Sommerfeld property. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8 (3), pp. 305–309.

Tolchennikov А.А., Chernyshev V.L., Shafarevich А.I. Schrödinger equation on graphs and singular spaces: spectral properties and semiclassical dynamics of localized packets, 2014, № 2, p. 75–104.

Yang Y., Zhao L. Normalized solutions for nonlinear Schrodinger equations on graphs, 2023, arXiv:2302.12585v1. Available: https://arxiv.org/abs/2302.12585

Ablowitz M., Musslimani Z. Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity, 2016, 29, pp. 915-946.

Aseeva N.V., Gromov E.M., Podchishaeva O.V., Tyutin V.V. Soliton dynamics in the frame of extended inhomogeneous nonlinear Schrödinger equation with taking into account nonlocal nonlinearity. Proceedings of R.E. Alekseev Nizhny Novgorod State Technical University, 2013, No. 1(98), pp. 51-64.

Sabirova K., Matrasulov D., Akramov M., Susanto H. Nonlocal Nonlinear Schrodinger Equation on Metric Graphs, 2021, arXiv:2111.03271v1. Available: https://arxiv.org/pdf/2111.03271

Gavrikov A.M. Investigating quantum graph variations as functions of time. Polytechnic Youth Journal, 2017, No. 11, pp. 1-13.

Rybakov M.A. A symbolic method for solving the initial boundary

value problem for an inhomogeneous continuum transfer equation on a graph. Modeling, Optimization and Information Technology. 2024;12(2).

Egorov S.I., Sapozhnikov D.A., Usatyuk V.S. Application of Bethe energy approximation to determine the numerical characteristics of codes on a graph structures. Mathematical Modeling and Computational Methods, 2025, No. 1, pp. 104-115.

Buslaev A.P., Tatashev A.G., Yashina M.V. On Properties of Solutions of a Class of Systems of Nonlinear Differential Equations on Graphs. Vladikavkaz Mathematical Journal, 2004, Vol. 6, No. 4, pp. 17-24.

Kholodov Ya. A. Development of network computational models for the study of nonlinear wave processes on graphs. COMPUTER RESEARCH AND MODELING, 2019 VOL. 11 NO. 5 P. 777–814.

Dimitrienko Yu.I., Ivanov M.Yu. Modeling of Nonlinear Dynamic Transport Processes in Porous Media. Herald of Bauman Moscow State Technical University, Series "Natural Sciences", 2008, No. 1, pp. 24-38.

Uvarova L. A., Linn P. W. Modeling of the “reaction–diffusion” transfer process in a nonlinear electromagnetic feld, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2021, vol. 25, no. 4, pp. 663–675. (In Russian).

Scott A. Nonlinear Science: Emergence and Dynamics of Coherent Structures. 2nd ed. Oxford: Oxford University Press, 2003, 559 p.


Refbacks

  • There are currently no refbacks.


Abava  Кибербезопасность Monetec 2026 СНЭ

ISSN: 2307-8162