Macrosystem dynamics
Д.А. Буров "Моделирование связанного нелинейного уравнения Шрёдингера конечно-разностными методами"
Intellectual systems and technologies
Information Technology
System analysis in medicine and biology
Д.А. Буров "Моделирование связанного нелинейного уравнения Шрёдингера конечно-разностными методами"


В работе освещается численный анализ связанного нелинейного уравнения Шрёдингера в приложении к моделированию поверхностных плазмон-поляритонов. Для решения системы уравнений в частных производных применялись конечно-разностные схемы высоких порядков, в том числе, с использованием схем Паде (компактных разностных производных) и методов Дормана-Принса. Основной акцент сделан на применимости конечно-разностных методов к данной задаче. Рассмотрены различные типы граничных условий (Дирихле, Неймана, периодические). Помимо этого, представлены результаты моделирования, изучено усложнение динамики при изменении одного из системных параметров. Сделан вывод о начальных стадиях перехода к хаотическим режимам.

Ключевые слова:

нелинейное уравнение Шрёдингера, уравнение Гинзбурга-Ландау, поверхностный плазмон-поляритон, метод Кранка-Николсона, компактные разности, схема Паде, устойчивость по фон Нейману, сценарий ФШМ, сценарий Ландау-Хопфа, трехмерный тор, субкритическая бифуркация Хопфа, мультистабильность.

Стр. 18-32.

Полная версия статьи в формате pdf. 

D.A. Burov

"Modelling of coupled nonlinear Schroedinger equation using finite difference methods"

Abstract. This paper presents numerical analysis of coupled nonlinear Schroedinger equation as an application to surface plasmon polariton modelling. High-order finite difference methods are used to solve a system of partial differential equations; methods include Pade schemes (compact finite differences) and Dormand-Prince method. Main emphasis is on the applicability of finite difference methods to this particular problem. Different types of boundary conditions are considered (Dirichlet, Neumann, periodic). Besides, simulation results are also presented, and the rise in dynamics complexity is studied. Initial stages of transition to chaos are discussed.

Keywords: nonlinear Schroedinger equation, Ginzburg-Landau equation, surface plasmon polariton, Crank-Nicolson method, compact finite difference, Pade scheme, von Neumann stability, FSM scenario, Landau-Hopf scenario, three-dimensional torus, subcritical Hopf bifurcation, multistability.


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