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    廣義Sasa-Satsuma 呈在半直線上的初邊值問題

    2019-10-28 02:19董鳳嬌胡貝貝

    董鳳嬌 胡貝貝

    摘要:本文基于Fokas統(tǒng)一變換方法分析了廣義Sasa-Satsuma方程在半直線上的初邊值問題.假設(shè)廣義S asa-Satsuma方程的解u(x,t)存在,證明了其初邊值問題的解可用復譜參數(shù)λ平面上的3×3矩陣Riemann-Hilbert問題的形式解唯一表示.

    關(guān)鍵詞:Riemann-Hilbert問題; 廣義Sasa-Satsuma方程; 初邊值問題; Fokas統(tǒng)一變換

    中圖分類號:0175.29

    文獻標志碼:A

    DOI: 10.3969/j.issn.1000-5641.2019.04.004

    0 引言

    自Gardner, Green,Kruskal,Miura發(fā)現(xiàn)了反散射變換以來,一直到20世紀90年代,反散射變換幾乎只是用來分析純初值問題,但是在現(xiàn)實自然界中,越來越多的自然現(xiàn)象需要考慮邊值條件,這樣就自然地需要考慮初邊值問題來取代初值問題.1997年,F(xiàn)okas[1]基于反散射變換的思想首次提出了統(tǒng)一變換方法,很好地求解了可積方程的初邊值問題.在過去的20年里,該方法已經(jīng)用來分析了一些具有2x2矩陣Lax對的重要可積方程的初邊值問題[2-5].就像全直線上的反散射方法一樣,F(xiàn)okas方法也是將初邊值問題的解表示成相應的Riemann-Hilbert問題的解.2012年,Lenells[6]首次將此方法推廣到3x3矩陣可積方程,并且研究了Degasperis-Procesi方程在半直線上的初邊值問題[7]在這之后,越來越多的學者開始關(guān)注Riemann-Hilbert問題,使得許多與高階矩陣譜問題相關(guān)的可積方程初邊值問題得以研究,比如,Novikov方程[8]、Sasa-Stsuma方程[9]、耦合NLS方程等[11-12],作者在這方面也做了一些工作[13-16].

    眾所周知,非線性薛定諤方程

    4 結(jié)論

    在本文中,我們構(gòu)造了一個新的雙模耦合KdV方程,一方面,通過簡化的Hirota方法和Cole-Hopf變換,對于特殊的α、β值可得到該方程的孤子解,但對于一般的α、β值,孤子解是否存在,我們還不能確定.另一方面,通過不同的函數(shù)展開法,對于一般的α、β值,我們得到了該方程的其他精確解.

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