Алгоритм построения первых интегралов одного класса нелинейных дифференциальных уравнений второго порядка

Рассматривается один из классов нелинейных обыкновенных дифференциальных уравнений второго порядка. Предлагается алгоритм построения первых интегралов рассматриваемого класса уравнений. Для поиска аналитических решений уравнений данного класса, как правило, используются специальные подходы типа метода простейших уравнений. Предлагаемый алгоритм позволяет в ряде случаев находить общие решения нелинейных дифференциальных уравнений. Применение алгоритма иллюстрируется на примере комплексного уравненния Гинзбурга–Ландау, решение которого ищется используя переменные бегущей волны. Показано, что нелинейное обыкновенное дифференциальное уравение второго порядка достаточно сложного вида, с помощью предлагаемого алгоритма можно свести к дифференцильному уравнению первого порядка, решение которого в общем виде можно представить в виде квадратуры. При нулевых значениях постоянных интегрирования, точные решения уравнения Гинзбурга–Ландау получены в виде аналитических выражениий. Представлены точные решения уравнения Гинзбурга–Ландау в виде периодических и уединенных волн, которые выражаются через эллиптические и гиперболические функции.

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