vestnikmephiВестник НИЯУ МИФИVestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI"2304-487XNational Research Nuclear University "MEPhI"10.26583/vestnik.2024.5.6PALOUNvestnikmephi-365Research ArticleМАТЕМАТИЧЕСКИЕ МОДЕЛИ И ЧИСЛЕННЫЕ МЕТОДЫMATHEMATICAL MODELS AND NUMERICAL METHODSНЕЛИНЕЙНЫЕ УРАВНЕНИЯ ШРЕДИНГЕРА С ЗАПАЗДЫВАНИЕМ: ТОЧНЫЕ РЕШЕНИЯ, РЕДУКЦИИ И ПРЕОБРАЗОВАНИЯNONLINEAR SCHRÖDINGER EQUATIONS WITH DELAY: EXACT SOLUTIONS, REDUCTIONS, AND TRANSFORMATIONShttps://orcid.org/0000-0002-2610-0590ПолянинА. Д.PolyaninA. D.

главный научный сотрудник

AuthorID: 4251

polyanin@ipmnet.ruhttps://orcid.org/0000-0001-5926-9715КудряшовН. А.KudryashovN. A.

главный научный сотрудник, заведующий кафедрой

доктор физико-математических наук, профессор 

nakudr@gmail.comИнститут проблем механики им. А.Ю. Ишлинского РАН; Национальный исследовательский ядерный университет «МИФИ»РоссияIshlinsky Institute for Problems in Mechanics, Russian Academy of SciencesRussian FederationНациональный исследовательский ядерный университет «МИФИ»РоссияNational Research Nuclear University MEPhI (Moscow Engineering Physics Institute)Russian Federation202405112024135340349Copyright © Полянин А.Д., Кудряшов Н.А., 20242024Полянин А.Д., Кудряшов Н.А.Polyanin A.D., Kudryashov N.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnikmephi.elpub.ru/jour/article/view/365

Рассматриваются уравнения Шредингера с кубическими и более сложными нелинейностями, содержащими искомую функцию с запаздывающим аргументом. Высказаны физические соображения о возможных причинах появления запаздывания в подобных нелинейных уравнениях и моделях. Описаны одномерные редукции, приводящие исследуемые уравнения в частных производных с запаздыванием к более простым обыкновенным дифференциальным уравнениям или обыкновенным дифференциальным уравнениям с запаздыванием. Найдены точные решения нелинейного уравнения Шредингера общего вида с запаздыванием, которые выражаются в квадратурах. Особое внимание уделено трем уравнениям специального вида с кубической нелинейностью, которые допускают простые решения в элементарных функциях, а также более сложные точные решения с обобщенным разделением переменных. Помимо нелинейных уравнений Шредингера с постоянным запаздыванием исследуются также некоторые более сложные уравнения с переменным запаздыванием общего вида. Полученные результаты могут быть полезны для тестирования математических моделей, описываемых нелинейными уравнениями Шредингера с запаздыванием и родственными уравнениями математической физики.

Schrödinger equations with cubic and more complex nonlinearities containing the desired function with a delayed argument are considered. The physical considerations that can lead to the appearance of a delay in such nonlinear equations and models are expressed. One-dimensional reductions are described that lead the studied partial differential equations with delay to simpler ordinary differential equations or ordinary differential equations with delay. Exact solutions of the nonlinear Schrödinger equation of general form with delay, which are expressed in quadratures, are found. Special attention is paid to three equations with cubic nonlinearity, which allow simple solutions in elementary functions, as well as more complex exact solutions with generalized separation of variables. In addition to nonlinear Schrödinger equations with constant delay, some more complex equations with variable delay of general form are also studied. The results obtained can be useful for testing mathematical models described by the nonlinear Schrödinger equation with delay.

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The authors declare that there are no conflicts of interest present.