Уравнение Кортевега – де Вриза – Бюргерса с нелинейным источником: редукция, тест Пенлеве, первые интегралы и аналитические решения

Изучается уравнение Кортевега – де Вриза – Бюргерса с нелинейным источником. Задача Коши для этого уравнения в общем случае не решается методом обратного преобразования рассеяния. Однако, уравнение допускает группу преобразований сдвига по независмым переменным и поэтому рассматривается с учетом переменных бегущей волны. Для исследования аналитических свойств нелинейного обыкновенного дифференциального уравнения применяются три шага теста Пенлеве. Показано, что в общем случае уравнение не проходит тест Пенлеве. Из анализа существования ряда Лорана для общего решения дифференциального уравнения получены условия на параметры математической модели при которых уравнение проходит тест Пенлеве и, следовательно, выполняются необходимые условия существования общего решения для четырех случаев нелинейного обыкновенного дифференциального уравнения. Принимая во внимание значения индексов Фукса найдены первый интеграл соответствующего нелинейного обыкновенного дифференцишльного уравнения. Показано, что общие решения одного из нелинейных обыкновенных  дифференциальных уравнений выражаются через эллиптическую функцию Вейерштрасса, а решения  другого уравнения имеет решение представимые через трансценденты первого уравнения Пенлеве при определенных параметрических ограничениях на параметры уравнения. Обсуждается взаимосвязь между тестом Пенлеве и специальными методами нахождения точных решений нелинейных дифференциальных уравнений. Специальные методы используются для построения аналитических решений с одной и двумя произвольными постоянными. Получены точные решения с двумя произвольными постоянными, выраженными через эллиптическую функцию Вейерштрасса. С помощью метода логистических функций найдены точные решения уравнения Кортевега-де Вриза-Бюргерса с нелинейным источником с одной произвольной постоянной. Показано, что семейство уравнений, для которых найдены точные решения, значительно расширяется в случае использования специальных методов.

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