Обратное преобразование рассеяния для дробного уравнения Кортевега — де Фриза пятого порядка с самосогласованным источником
DOI:
https://doi.org/10.56143/ujmcs.v2i3s.6Ключевые слова:
Riesz fractional Korteweg-de Vries equation, self-consistent source, inverse scattering transform method, Sturm-Liouville operatorАннотация
В настоящей работе мы строим и интегрируем дробное уравнение Кортевега — де Фриза пятого порядка с производной Рисса и самосогласованным источником (fRfKdV) в классе быстро убывающих функций с помощью метода обратного преобразования рассеяния (IST). Показано, что дробное уравнение Кортевега — де Фриза пятого порядка с самосогласованными источниками представляет собой важную теоретическую модель, поскольку является вполне интегрируемой системой. Для одно-солитонного случая получены явные формулы решений рассматриваемой задачи. На их основе при различных значениях дробного порядка моделируются пространственные профили решений и исследуются их структурные свойства. Полученные результаты показывают, что как источник, так и дробный порядок существенно влияют на скорость распространения солитона, при этом на протяжении всего процесса движения диссипация энергии отсутствует. Такое поведение соответствует сверхдисперсионному механизму переноса и представляет собой экспериментально проверяемое предсказание настоящей теории. Кроме того, проведённый анализ показывает, что с увеличением параметра $\epsilon$ абсолютное значение скорости волны возрастает. Настоящее исследование даёт ценные сведения о нелинейной динамике и поведении солитонов, способствуя более глубокому пониманию многообразных волновых взаимодействий в нелинейных дробных уравнениях.
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