Defocusing Nonlocal Nonlinear Schrödinger Equation with Step-like Boundary Conditions: Long-time Behavior for Shifted Initial Data
DOI:
https://doi.org/10.15407/mag16.04.418Анотація
Стаття присвячена асимптотиці за великим часом початкової задачі для інтегровного дефокусуючого нелокального нелінійного рівняння Шредінгера $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ з початковими даними типу сходинки: $q(x,0)\to 0$ при $x\to -\infty$ та $q(x,0)\to A$ при $x\to +\infty$. Через те, що це рівняння не є трансляційно інваріантним, розв'язок цієї задачі чутливий до зміщень початкових даних. Ми розглядаємо сім'ю задач, параметризованих параметром $R>0$, з початковим даними, які можуть розглядатися як збурення "зміщеної сходинки" $q_{R,A}(x)$: $q_{R,A}(x)=0$ для $x<R$ та $q_{R,A}(x)=A$ для $x>R$, де $A>0$ та $R>0$ є довільними константами. Ми показуємо, що асимптотика розв'язку задачі за великим часом якісно різна у секторах $(x,t)$ площини, кількість яких залежить від значень $A$ та $R$: для фіксованого $A$, чим більше $R$, тим більша кількість секторів.
Mathematics Subject Classification: 35B40, 35Q15, 35B30
Ключові слова:
нелокальне нелінійне рівняння Шредінгера, задача Рімана-Гільберта, асимптотика за великим часом, нелінійний метод перевалуПосилання
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