汪春江,舒 級,李 倩,王云肖,楊 袁

(四川師范大學(xué) 數(shù)學(xué)與軟件科學(xué)學(xué)院,四川 成都 610066)

一類(3+1)維KdV方程的有理解及其怪波

汪春江,舒 級*,李 倩,王云肖,楊 袁

(四川師范大學(xué) 數(shù)學(xué)與軟件科學(xué)學(xué)院,四川 成都 610066)

討論一類經(jīng)典的(3+1)維KdV方程,該方程在流體動力學(xué)、等離子物理、氣體動力學(xué)等方面有廣泛應(yīng)用.通過一個簡單的符號計算方法得到方程的有理解,并討論了在某些條件下的怪波解.

KdV方程; 精確解; 符號計算方法; 有理解; 怪波

一直以來,非線性現(xiàn)象都是基礎(chǔ)數(shù)學(xué)和應(yīng)用數(shù)學(xué)研究的主題,非線性演化方程的精確解研究在數(shù)學(xué)物理上有著重大作用,常系數(shù)方程[1-3]、變系數(shù)方程[4-5]、隨機方程[6-7]的精確解已經(jīng)被廣泛研究,并產(chǎn)生了很多關(guān)于精確解的方法,如逆散映射法[8-9]、Backlünd變換[10-11]、達(dá)布變換[12]、齊次平衡法[13]、(G′/G)-展開法[14]等.

近年來,怪波已成為國內(nèi)外研究的焦點.從怪波解的形式上看,通常是有理分式.在海洋學(xué)和其他學(xué)科領(lǐng)域,科學(xué)家們都發(fā)現(xiàn)了怪波現(xiàn)象[15-17],例如,Bose-Einstein凝聚物怪波事件[18-19],在等離子體的空間和表面的異常波[20-21],尤其是在光學(xué)領(lǐng)域[22-23],當(dāng)光脈沖在光子晶體纖維中傳輸高能量時,怪波就會存在.對于非線性Schr?dinger方程[24-25]的怪波與駐波、多怪波和高階怪波[26-28]、明暗怪波解[29-30]已廣泛被討論.怪波不僅出現(xiàn)在深水中,淺水中也發(fā)現(xiàn)了怪波[31].從直觀上看,怪波具有超常的波高,因此大多數(shù)學(xué)者和研究人員只能從波高角度對它進(jìn)行定義,即認(rèn)為波高大于有效波高2倍(或2.2倍)的單波可以稱為怪波.在淺水中,怪波的產(chǎn)生取決于調(diào)制不穩(wěn)定性:當(dāng)波高kH<1.363 m[32],非線性聚焦過程停止.以水為介質(zhì)的波不同于一般的閾值,淺水波高為kH<1/3 m[33].

在本文中,考慮(3+1)維KdV方程

(1)

其中u是關(guān)于x、y、z、t的函數(shù),x、y、z、t是獨立變量.它是物理學(xué)家和數(shù)學(xué)家感興趣的方程之一.KdV方程的復(fù)合解[35]、變系數(shù)KdV方程的精確解[36]、高階KdV方程的精確解[37-39]、新的KdV-mKdV方程的孤波[40]、廣義的Hirota Satsuma耦合方程和偶合的MKdV方程[41-42]的孤立波解已經(jīng)被研究.

本文的目的是通過一個簡單的符號計算方法[43],構(gòu)建出(3+1)維KdV的怪波和有理解.

1 一個簡單的符號計算方法

一個簡單的符號計算方法對于求解非線性偏微分方程是有效的.下面給出求解(3+1)維偏微分方程的主要步驟.

步驟 1 利用截斷展開法[44-46],作代換

(2)

其中,α是一個常數(shù),j1,j2,j3,j4≥1(j1,j2,j3,j4∈0,1,2,…).通過上述變換將一個(3+1)維的偏微分方程

(3)

轉(zhuǎn)化為一個雙線性方程

(4)

其中F是f,ft,fx,fy,fz,fxx,fxy,fxz,…的多項式.

步驟 2 假設(shè)f是一個關(guān)于x、y、z、t的2N階多項式,給定

(5)

其中,系數(shù)ai,j,k,l(0≤i,j,k,l≤2N)是常數(shù),滿足

(6)

如果j1≥1.

步驟 3 把(5)式代入(4)式,并令關(guān)于xiyjzktl(0≤i+j+k+l≤2N)的系數(shù)為0,得到一系列的多項式方程.

步驟 4 利用Maple軟件,可以算出系數(shù)aijkl,i,j,k,l∈(0,1,2,…).

步驟 5 把滿足條件(6)的系數(shù)aijkl(i,j,k,l∈{0,1,2,…})代入(2)式之后得到非線性偏微分方程(3)的解.

2 有理解

下面將應(yīng)用上述方法來求解(3+1)維KdV方程的解.根據(jù)截斷展開法,用變換

(7)

將(3+1)維KdV方程變?yōu)?/p>

令f成為上述方程的一個解

(8)

為了構(gòu)建方程(1)的有理解,假定f是一個2階關(guān)于x、y、z、t的多項式

(9)

其中系數(shù)aijkl(0≤i,j,k,l≤2)是常數(shù),滿足

(10)

為了簡單起見,假定方程(9),a2000=a0200=a0020=a0000=1,a0002=2,代入到方程(8)中,并令關(guān)于xiyjzktl(0≤i+j+k+l≤2)的系數(shù)為0,得到15個多項式方程:

為了后面描述的方便,令a0011=b,a1100=c,a1001=d,a0110=e,a0101=f,a1010=g,a0001=h,a0010=j,a0100=k,a1000=l,解方程組,得到如下解.

情形 1

f=-ge,

情形 2

f=-ge,

情形 3

f=-ge,

情形 4

f=-ge,

情形 5

情形 6

情形 7

情形 8

情形 9

情形 10

情形 11

情形 12

情形 13

情形1～13代入(7)式得到(3+1)維KdV方程的解,情形1～4有2個自由變量a1100、a0010.在這些情形中可以得到實值型的怪波解.情形5～12有一個自由變量a0010.在這些情況中得到復(fù)值型的有理解.然而,在情形13中,有一個自由變量a0010.在這個情形中,可以得到實值型的有理解.

1) 從情形1中,挑選出系數(shù):

可以得到(3+1)維KdV方程的實值型怪波解

其中

(12)

2) 從情形5中,挑選出系數(shù):

可以得到(3+1)維KdV方程的復(fù)值型有理解

(13)

其中

(14)

3) 從情形13中,挑選出系數(shù):

可以得到(3+1)維KdV方程的實值型有理解

其中

(16)

3 結(jié)束語

本文運用符號計算方法得到了(3+1)維KdV方程的有理解和怪波解,這些怪波和有理解是非奇異的.這些解對理解怪波的產(chǎn)生機制有一定幫助.下一步將研究如何用簡單的符號計算方法構(gòu)造非線性演化方程的高階怪波解.

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2010 MSC：35Q55

(編輯 周 俊)

Rogue Waves and Rational Solutions for a Class of (3+1)-dimensional KdV Equation

WANG Chunjiang,SHU Ji,LI Qian,WANG Yunxiao,YANG Yuan

(CollegeofMathematicsandSoftwareScience,SichuanNormalUniversity,Chengdu610066,Sichuan)

This paper discusses a classical (3+1)-dimensional KdV equation,which has broad applications in hydrodynamics,plasma physics,gas dynamics.We obtain rational solutions of this equation by a simple symbolic computation approach.Under some conditions,we find that some of rational solutions are rogue waves.

KdV equation; exact solution; symbolic computation approach; rational solution; rogue wave

2016-03-30

四川省科技廳應(yīng)用基礎(chǔ)計劃項目(2016JY0204)和四川省教育廳自然科學(xué)重點基金(14ZA0031)

O175.27

A

1001-8395(2017)02-0157-06

10.3969/j.issn.1001-8395.2017.02.003

*通信作者簡介：舒 級(1976—),男,教授,主要從事隨機動力系統(tǒng)和偏微分方程的研究,E-mail:shuji2008@hotmail.com