帶參數(shù)時(shí)空分?jǐn)?shù)階Fokas-Lenells 方程的精確解*

劉楊秀， 胡彥霞

（華北電力大學(xué) 數(shù)理學(xué)院，北京 102206）

引 言

近幾十年來(lái)，分?jǐn)?shù)階非線性偏微分方程的研究已滲透到許多科學(xué)領(lǐng)域中.由整數(shù)階微分方程推廣出的分?jǐn)?shù)階微分方程，在描述一些系統(tǒng)的動(dòng)力學(xué)行為中，更加能反映出系統(tǒng)的實(shí)際變化規(guī)律[1].眾所周知的分?jǐn)?shù)階非線性Schr?dinger 方程就是非常典型的一類非線性發(fā)展方程.很多分?jǐn)?shù)階非線性偏微分方程與深水波動(dòng)、潛水波動(dòng)現(xiàn)象有著緊密的聯(lián)系.如果知道這類方程的精確解，將有利于數(shù)值模擬進(jìn)行檢驗(yàn)以及定性分析.由此我們需要了解分?jǐn)?shù)階導(dǎo)數(shù)各種詳細(xì)的定義，比如R-L 分?jǐn)?shù)階導(dǎo)數(shù)、Caputo 分?jǐn)?shù)階導(dǎo)數(shù)等[2-4]，同樣用來(lái)解決整數(shù)階非線性偏微分方程的方法，也可以應(yīng)用到分?jǐn)?shù)階非線性偏微分方程上.從現(xiàn)有文獻(xiàn)看，求解分?jǐn)?shù)階非線性偏微分方程的方法已有眾多，比如試探函數(shù)法[5]、Horita 法[6]、擴(kuò)展的直接代數(shù)法[7]、李群方法[8-9]、Jacobi 橢圓函數(shù)法[10]、擴(kuò)展的Jacobi 橢圓函數(shù)法[11]、廣義G′/G方法[12]、sine-Gordon 方法[13]、多項(xiàng)式完全判別系統(tǒng)法[14]以及其他求解方法[15-24].由于非線性分?jǐn)?shù)階偏微分方程的多樣性及復(fù)雜性和每一種求解方法的局限性，還未能有一種通用的普遍有效的求解方法.因此，運(yùn)用有效的方法去求非線性分?jǐn)?shù)階偏微分方程的精確解仍是一個(gè)需要不斷研究的課題之一.隨著MATLAB 等其他數(shù)學(xué)軟件的應(yīng)用和普及，借助計(jì)算機(jī)軟件可以幫助我們更方便地解決這一問(wèn)題.

本文主要考慮非線性光學(xué)領(lǐng)域中的帶參數(shù)時(shí)空分?jǐn)?shù)階Fokas-Lenells 方程[25]：

多項(xiàng)式完全判別系統(tǒng)法是一種求解此類問(wèn)題比較有效的方法，是將偏微分方程在行波變換下簡(jiǎn)化為常微分方程，再對(duì)常微分方程中的多項(xiàng)式進(jìn)行完整分類并求解相應(yīng)積分，從而得到原方程的精確解.對(duì)于分?jǐn)?shù)階偏微分方程，若其在行波變換下簡(jiǎn)化為u′(ξ)=G(u,θ1,θ2,···,θm)(θ1,θ2,···,θm為 相應(yīng)參數(shù)，且G(u,θ1,θ2,···,θm)是關(guān)于u的多項(xiàng)式)的形式，就可以利用此方法進(jìn)行求解.關(guān)于此方法的介紹及應(yīng)用詳見(jiàn)文獻(xiàn)[37-41].

本文考慮方程(1)在一般情況下的解的問(wèn)題，利用多項(xiàng)式完全判別系統(tǒng)法，根據(jù)對(duì)方程(1)單行波解的完整分類，在不做任何參數(shù)限制的條件下，求得方程(1)在一般情況下的9 種精確解，包括有理函數(shù)解、周期解、孤波解、Jacobi 橢圓函數(shù)解和雙曲函數(shù)解等，繪制了精確解的相關(guān)圖像，由此分析了參數(shù)對(duì)解的結(jié)構(gòu)的影響.

1 預(yù) 備 知 識(shí)

2 帶參數(shù)時(shí)空分?jǐn)?shù)階Fokas-Lenells 方程的約化分析

首先對(duì)方程進(jìn)行行波變換，令

3 帶參數(shù)Fokas-Lenells 方程的精確解

圖1 | Ω1(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖1(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω1(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω1(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω1(x,t)|2 的 三維圖； (c) 圖1(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω1(x,t)|2關(guān) 于x 的截面圖Fig. 1 The 3D graph of | Ω1(x,t)|2 with different values of the fractional derivative, the contour plot of fig.1(b) and the sectional view of | Ω1(x,t)|2 against x with t =3: (a) the graphic model of | Ω1(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω1(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.1(b);(d) the sectional view of | Ω1(x,t)|2 against x when t=3

圖2 | Ω2(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖2(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω2(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω2(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω2(x,t)|2 的 三維圖； (c) 圖2(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω2(x,t)|2關(guān) 于x 的截面圖Fig. 2 The 3D graph of | Ω2(x,t)|2 with different values of the fractional derivative, the contour plot of fig.2(b) and the sectional view of | Ω2(x,t)|2 against x with t =3: (a) the graphic model of | Ω2(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω2(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.2(b);(d) the sectional view of | Ω2(x,t)|2 against x when t=3

經(jīng)檢驗(yàn)，此情況不成立，不予討論.

圖3 | Ω3(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖3(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω3(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω3(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω3(x,t)|2 的 三維圖； (c) 圖3(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω3(x,t)|2關(guān) 于x 的截面圖Fig. 3 The 3D graph of | Ω3(x,t)|2 with different values of the fractional derivative, the contour map of fig. 3(b) and the sectional view of | Ω3(x,t)|2 against x with t =3: (a) the graphic model of | Ω3(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω3(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 3(b);(d) the sectional view of | Ω3(x,t)|2 against x when t=3

圖4 | Ω4(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖4(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω4(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω4(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω4(x,t)|2 的 三維圖； (c) 圖4(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω4(x,t)|2關(guān) 于x 的截面圖Fig. 4 The 3D graph of | Ω4(x,t)|2 with different values of the fractional derivative, the contour plot of fig.4(b) and the sectional view of | Ω4(x,t)|2 against x with t =3: (a) the graphic model of | Ω4(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω4(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.4(b);(d) the sectional view of | Ω4(x,t)|2 against x when t=3

圖5 | Ω5(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖5(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω5(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω5(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω5(x,t)|2 的 三維圖； (c) 圖5(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω5(x,t)|2關(guān) 于x 的截面圖Fig. 5 The 3D graph of | Ω5(x,t)|2 with different values of the fractional derivative, the contour plot of fig. 5(b) and the sectional view of | Ω5(x,t)|2 against x with t =3: (a) the graphic model of | Ω5(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω5(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 5(b);(d) the sectional view of | Ω5(x,t)|2 against x when t=3

圖6 | Ω6(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖6(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω6(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω6(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω6(x,t)|2 的 三維圖； (c) 圖6(b)的等高線圖；(d) 當(dāng)t =3 時(shí)| Ω6(x,t)|2關(guān) 于x 的截面圖Fig. 6 The 3D graph of | Ω6(x,t)|2 with different values of the fractional derivative, the contour plot of fig. 6(b) and the sectional view of | Ω6(x,t)|2 against x with t =3: (a) the graphic model of | Ω6(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω6(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 6(b);(d) the sectional view of | Ω6(x,t)|2 against x when t=3

圖7 | Ω7(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖7(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω7(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω7(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω7(x,t)|2 的 三維圖； (c) 圖7(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω7(x,t)|2關(guān) 于x 的截面圖Fig. 7 The 3D graph of | Ω7(x,t)|2 with different values of the fractional derivative, the contour plot of fig.7(b) and the sectional view of | Ω7(x,t)|2 against x with t =3: (a) the graphic model of | Ω7(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω7(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.7(b); (d) the sectional view of | Ω7(x,t)|2 against x when t=3

圖8 | Ω8(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖8(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω8(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω8(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω8(x,t)|2 的 三維圖； (c) 圖8(b)的等高線圖； (d) 當(dāng)t =3 時(shí)α =1/2,β=1/3,|Ω8(x,t)|2 關(guān) 于x 的截面圖Fig. 8 The 3D graph of | Ω8(x,t)|2 with different values of the fractional derivative, the contour plot of fig. 8(b) and the sectional view of | Ω8(x,t)|2 against x with t =3: (a) the graphic model of | Ω8(x,t)|2,α=1/2,β=1/3; (b) the graphic model of |Ω 8(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig. 8(b);(d) the sectional view of | Ω8(x,t)|2 against x when t=3

圖9 | Ω9(x,t)|2取 不同分?jǐn)?shù)階導(dǎo)數(shù)值時(shí)的三維圖，圖9(b)對(duì)應(yīng)的等高線圖，以及t =3 時(shí)| Ω9(x,t)|2關(guān) 于 x 的 截面圖：(a) α =1/2,β=1/3,|Ω9(x,t)|2的三維圖； (b) α =1/3,β=1/3,|Ω9(x,t)|2 的 三維圖； (c) 圖9(b)的等高線圖； (d) 當(dāng)t =3 時(shí)| Ω9(x,t)|2關(guān) 于x 的截面圖Fig. 9 The 3D graph of | Ω9(x,t)|2 with different values of the fractional derivative, the contour plot of fig.9(b) and the sectional view of | Ω9(x,t)|2 against x with t =3: (a) the graphic model of | Ω9(x,t)|2,α=1/2,β=1/3; (b) the graphic model of | Ω9(x,t)|2,α=1/3,β=1/3; (c) the contour plot of fig.9(b);(d) the sectional view of | Ω9(x,t)|2 against x when t=3

4 結(jié) 果 分 析

方程精確解的結(jié)構(gòu)反映了光學(xué)系統(tǒng)描述的波在介質(zhì)中傳播的特性.這里，我們主要討論分析分?jǐn)?shù)階參數(shù)α ， β 的變化對(duì)解的結(jié)構(gòu)的影響.根據(jù)前面對(duì)方程(1)精確解的 |Ωi(x,t)|2的相關(guān)圖形的分析發(fā)現(xiàn)，在圖1、3、5、7、9 中，當(dāng)其中一參數(shù) β不變，參數(shù)α 變化時(shí)，對(duì)應(yīng)的方程的奇異解的結(jié)構(gòu)沒(méi)有發(fā)生本質(zhì)的變化，只有波峰會(huì)向左或向右偏移，或出現(xiàn)幾個(gè)零散的波峰.由于這些奇異解在波峰處有一個(gè)不連續(xù)的一階導(dǎo)數(shù)，這反映出對(duì)應(yīng)的方程(1)描述的波傳播的特性.在圖2、8 中，當(dāng)參數(shù)α 變化時(shí)，奇異周期解的結(jié)構(gòu)及奇異性、周期性也并未發(fā)生大的改變.在圖4 中，隨著 α的減小，扭波峰值會(huì)向右移動(dòng)，但扭波仍然存在，其等高線分布僅在很小的時(shí)間范圍內(nèi)波動(dòng)，但在t=0.1左 右出現(xiàn)新的走勢(shì).在圖6 中，隨著α 的變化，對(duì)應(yīng)的奇異解的結(jié)構(gòu)也沒(méi)有發(fā)生質(zhì)的變化，但從等高線密集程度來(lái)看，有幾條明顯的“山脈”，反映了此時(shí)方程(1)描述的波傳播的情況.

5 結(jié) 論

本文通過(guò)行波變換，將光學(xué)系統(tǒng)中帶參數(shù)時(shí)空分?jǐn)?shù)階Fokas-Lenells 方程轉(zhuǎn)換成常微分方程，然后利用多項(xiàng)式完全判別系統(tǒng)法對(duì)該方程進(jìn)行了單行波解的完整分類，在不對(duì)方程中的參數(shù)和n做任何限定的情況下，得到了方程在一般情況下的精確解，包括有理函數(shù)解、孤立波解、雙曲函數(shù)解、周期解、Jacobi 橢圓函數(shù)解等.在現(xiàn)有文獻(xiàn)中，還沒(méi)有見(jiàn)到對(duì)帶參數(shù)時(shí)空分?jǐn)?shù)階Fokas-Lenells 方程中的參數(shù)和n不做任何限定的情況下求的精確解的相關(guān)結(jié)論.這是本文與其他已有文獻(xiàn)不同的地方.為了更好地理解此模型的物理現(xiàn)象和研究光孤子的傳播特性，我們繪制了精確解的相關(guān)三維圖、等高線圖及截面圖，討論了分?jǐn)?shù)階參數(shù)對(duì)解的影響.隨著分?jǐn)?shù)階導(dǎo)數(shù)值的變化，比如 β 不變， α減小時(shí)，方程的解的結(jié)構(gòu)并未發(fā)生質(zhì)的變化，這也反映出了此時(shí)光脈沖在介質(zhì)中的傳播特性.多項(xiàng)式完全判別系統(tǒng)法不僅可以用來(lái)求偏微分方程的精確解，也可以對(duì)方程進(jìn)行定性分析，這也將是我們此后工作的一部分.

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