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 數學與軟件科學學院,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 數學與軟件科學學院,四川 成都 610066)

討論一類經典的(3+1)維KdV方程,該方程在流體動力學、等離子物理、氣體動力學等方面有廣泛應用.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,非線性現象都是基礎數學和應用數學研究的主題,非線性演化方程的精確解研究在數學物理上有著重大作用,常系數方程[1-3]、變系數方程[4-5]、隨機方程[6-7]的精確解已經被廣泛研究,并產生了很多關于精確解的方法,如逆散映射法[8-9]、Backlünd變換[10-11]、達布變換[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.在海洋學和其他學科領域,科學家們都發現了怪波現象[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],尤其是在光學領域[22-23],當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.怪波不僅出現在深水中,淺水中也發現了怪波[31].從直觀上看,怪波具有超常的波高,因此大多數學者和研究人員只能從波高角度對它進行定義,即認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,怪波的產生取決于調制不穩定性:當波高kH<1.363 m[32],非線性聚焦過程停止.以水為介質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是關于x、y、z、t的函數,x、y、z、t是獨立變量.它是物理學家和數學家感興趣的方程之一.KdV方程的復合解[35]、變系數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]的孤立波解已經被研究.

本文的目的是通過一個簡單的符號計算方法[43],構建出(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,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f是一個關于x、y、z、t的2N階多項式,給定

(5)

其中,系數ai,j,k,l(0≤i,j,k,l≤2N)是常數,滿足

(6)

如果j1≥1.

步驟 3 把(5)式代入(4)式,并令關于xiyjzktl(0≤i+j+k+l≤2N)的系數為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aijkl,i,j,k,l∈(0,1,2,…).

步驟 5 把滿足條件(6)的系數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

下面將應用上述方法來求解(3+1)維KdV方程的解.根據截斷展開法,用變換

(7)

將(3+1)維KdV方程變為

令f成為上述方程的一個解

(8)

為了構建方程(1)的有理解,假定f是一個2階關于x、y、z、t的多項式

(9)

其中系數aijkl(0≤i,j,k,l≤2)是常數,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,并令關于xiyjzktl(0≤i+j+k+l≤2)的系數為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:

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:

可以得到(3+1)維KdV方程的復值型有理解

(13)

其中

(14)

3) 從情形13中,挑選出系數:

可以得到(3+1)維KdV方程的實值型有理解

其中

(16)

3 結束語

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造非線性演化方程的高階怪波解.

[1] JEFFREY A.Exact solutions tp the KdV-Burgers’ equation[J].Wave Motion,1991,14:369-375.

[2] KHALFALLAH M.New exact traveling wave solution of the (3+1) dimensional Kadomtsev-Petviashvili(KP) equation[J].Commun Nonlinear Sci Numer Simulat,2009,14:1169-1175.

[3] MALLORY K,VAN GORDER R A,VAJRAVELU K.Several classes of exact solutions to the (1+1) Born-Infeld equation[J].Commun Nonlinear Sci Numer Simulat,2014,19:1669-1674.

[4] ABDEL LATIF M S.Some exact solutions of KdV equation with variable coeffcients[J].Commun Nonlinear Sic Numer Simulat,2011,16:1783-1786.

[5] LI J B.Exact traveling wave solution and dynamical behavior for the (n+1)-dimensional multiple sine-Gordon equation[J].Science in China,2007,A50:153-164.

[6] KIM H S.Exact solutions of Wick-Type stochastic equations with variable coefficients[J].Reports on Mathematical Physics,2011,67:415-429.

[7] XIE Y.Exact solutions for stochastic KdV equations[J].Phys Lett,2003,A310:161-167.

[8] HEREMAN W,NUSEIR A.Symbolic methods to construct exact solutions of nonlinear partial differential equations[J].Math Comput Simul,1997,43:13-27.

[9] SACKS P,SHIN J.Computational methods for some inverse scattering problems[J].Appl Math Comput,2009,207:111-123.

[10] MA W X,ABDELJABBAR A.A bilinear Backlund transformation of a (3+1)-dimensional generalized KP equation[J].Applied Mathematics Letters,2012,25:1500-1504.

[11] QUINTINO A.Spectral deformation and Backlund transformation of constrained Willmore surfaces[J].Differential Geometry and its Applications,2011,29:S261-S270.

[12] BAGROV V G,SAMSONOV B F.Darboux transformation of the Schr?dinger equation[J].Physics of Particles and Nuclei,1997,28:374-397.

[13] ESLAMI M,FATHIVAJARGAH B.Exact solutions of modified Zakharov-Kuznetsov equation by the homogeneous balance method[J].Ain Shams Engineering J,2014,5:221-225.

[14] WANG M,LI X,ZHANG J.The (G′/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics[J].Phys Lett,2008,A372:417-423.

[15] DYSTHE K,KROGSTAD H E.Oceanic rogue waves[J].Annu Rev Fluid Mech,2008,40:287-310.

[16] KIM H S.Exact solutions of Wick-type stochastic equation with variable coefficients[J].Reports on Mathematical Physics,2011,67:415-429.

[17] PELINOVSKY K E,SLUNYAEV A.Rogue Waves in the Ocean: Observation,Theories and Modeling[M].New York:Springer,2009.

[18] LIA J Z,LIU H Y,WANG Q.Enhanced multilevel linear sampling methods for inverse scattering problems[J].J Comput Phys,2014,257:554-571.

[19] SACKS P,SHIN J.Computatioal methods for some inverse scattering problems[J].Appl Math Comput,2009,207:111-123.

[20] RUDERMAN M S.Freak waves in laboratory and space plasmas[J].Eur Phys J:Special Topics,2010,185:57-66.

[21] MOSLEM W M,SHUKLA P K,ELIASSON B.Surface plasma rogue waves[J].Eur Phys Lett,2011,96:25002.

[22] SOLLI D R,ROPERS C,KOONATH P,et al.Optical rogue waves[J].Nature,2007,450:1054-1058.

[23] KIBLER B,FATOME J,FINOT C.The Peregrine soliton in nonlinear fibre optics[J].Nature Physics,2010,6:790-795.

[24] SHU J,ZHANG J.Sharp criterion of global existence for a class of nonlinear Schr?dinger equation with critical exponent[J].Appl Math Comput,2006,182:1482-1487.

[25] HE J S,CHARALAMPADIS E G,KEVREKIDIS P G.Rogue waves in nonlinear Schr?dinger models with variable coefficients: application to Bose-Einstein condensates[J].Phys Lett,2014,A378:557-583.

[26] YU F J.Multi-rogue waves for a higher-order nonlinear Schrodinger equation in optical fibers[J].Appl Math Comput,2013,220:176-184.

[27] DAI C Q,HUANG W H.Multi-rogue wave and multi-breather solutions in PT-symmetric coupled waveguides[J].Applied Mathematics Letters,2014,32:35-40.

[28] LI L J,WU Z W,WANG L H,et al.High-order rogue waves for the Hirota equatoin[J].Ann Phys,2013,343:198-211.

[29] YU F J,YAN Z Y.New rogue waves and dark-bright soliton solutions for a coupled nonlinear Schr?dinger equation with variable coefficients[J].Appl Math Comput,2014,233:351-358.

[30] MA W X.Complexiton solutions to the Korteweg-de Vries equation[J].Phys Lett,2002,A301:35-44.

[31] SOOMERE T.Rogue waves in shallow water[J].Eur Phys J:Special Topics,2010,185:81-96.

[32] PETERSON P,SOOMERE T,ENGELBRECHT J.Soliton interaction as apossible model for extreme waves in shallow water[J].Nonlinear Processes in Geophysics,2003,10:503-510.

[33] RUDERMAN M S.Freak waves in laboratory and space plasmas[J].Eur Phys J:Special Topics,2010,185:57-66.

[34] BOUNTIS T C.Singularities and Dynamical Systems[M].New York:North-Holland,1985.

[35] MA H C,DENG A P,WANG Y.Exact solution of a KdV equation with variable coefficients[J].Appl Math Comput,2011,61:2278-2280.

[36] TRIKI H,WAZWAZ A-M.Traveling wave solutions for 5th-order KdV type equations with time-dependent coefficients[J].Commun Nonlinear Sic Numer Simulat,2014,19:404-408.

[37] OSBORNE A R,ONORATO M,SERIO M.The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains[J].Physics Letters,2000,A275:386-393.

[38] NOBUHIT M.Statistical properties of freak waves observed in the sea of Japan[C]//Proc International of Shore and Polar Engineering Conference,2000,3:109-115.

[39] DAVIDL K.Simulation of extreme waves in a background random sea[C]//Proc International of Shore and Polar Engineering Conference,2000,3:31-37.

[40] TAKASHI Y.Characteristics of giant freak waves observed in the sea of Japan[C]//Proc International 5-Symposium on Ocean Wave Measurement and Analysis,1997,1:316-328.

[41] HAVER S.A possible freak wave event measured at the draupner,jacket January 1 1995[C]//Actes de colloques-IFREMER,2004:1-8.

[42] OSBORNE A R,ONORATO M,SERIO M.The nonlinear dynamics of rogue waves and holes in deep water gravity wave trains[J].Physics Letters,2000,A275:386-393.

[43] OSBORNE A R.The random and deterministic dynamics of “rogue waves” in unidirectional deep-water wave trains[J].Marine Structures,2001,14:275-293.

[44] SLUNYAEV A,KHARIF C,PELINOVSKY E.Nonlinear wave focusing on water of finite depth[J].Physica D,2002,173:77-96.

[45] DYSTHE K,KROGSTAD H E,MULLER P.Oceanic rogue waves[J].Annu Rev Fluid Mech,2008,40:287-310.

[46] FAN E G.Solitons for a generalized Hirota-Satsuma coupled KdV equatioan and a coupled MKdV equation[J].Phys Lett,2001,A282:18-22.

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

四川省科技廳應用基礎計劃項目(2016JY0204)和四川省教育廳自然科學重點基金(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和偏微分方程的研究,E-mail:shuji2008@hotmail.com