Long-Time Asymptotics for the Modified Camassa–Holm Equation with Nonzero Boundary Conditions
DOI:
https://doi.org/10.15407/mag18.02.224Анотація
Ми розглядаємо модифiковане рiвняння Камаси–Хольма (мКХ) mt +(( u2 - ux2 )m)x =0, m := u − uxx на осi −∞ < x < +∞, де u(x,t) задовiльняє ненульовi крайовi умови на нескiнченностi: u(x,t) → 1 при x → ±∞. Метою роботи є дослiдження асимптотики за великим часом розв’язкiв початкової задачi, застосовуючи формалiзм задачi Рiмана–Гiльберта, що був нещодавно розроблений у [3]. Основна увага придiляється одержанню асимптотики у двох секторах пiвплощини (x,t) (t > 0), де основнi асимптотичнi члени мають вигляд модульованих та спадаючих (як t−1/2 ) тригонометричних осциляцiй, а також асимптотицi у секторi, де у поведiнцi розв’язку початкової задачi домiнують солiтони.
Mathematical Subject Classification 2010: 35Q53, 37K15, 35Q15, 35B40,
35Q51, 37K40
Ключові слова:
задача Рiмана–Гiльберта, нелiнiйний метод найшвидшого спуску, солiтониПосилання
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