摘要
高维非线性系统的非线性激发模式、复杂非线性波的研究是当前非线性孤子理论、波动理论和非线性动力学研究的重要课题。自然界中存在的各种非线性折叠现象,周期波形态需要我们从非线性波动模型中寻求更合理的理论诠释。本论文主要以源于实际水波和其他物理问题的2+1维非线性波动模型(方程)为研究对象,深入研究了它们的求解问题,揭示出丰富的非线性折叠孤波和周期传播波模式,并运用解析和图示方法对其非线性特性进行了探讨。本论文的主要工作及其研究成果可以分为三部分概述如下:
第一部分,本文将近年来发展的多线性分离变量方法、Painlevé截断展开法,广义的形变映射方法等分别推广应用到多个2+1维非线性波动模型,如2+1维长短波共振相互作用方程、Maccari系统、广义NNV方程、2+1维KdV方程、HBK方程,2+1维变系数BK方程等,获得他们的分离变量解.基于2+1维非线性分离变量解,选择适当的单值和多值函数,获得了丰富的非线性单值和多值局域激发模式,特别是多值折叠孤立波激发。
在2+1维分离变量解中选择适当的的单值分离变量函数,构造了2+1维非线性波动模型多种单值相干孤子结构如Dromion,Dromion格子,Solitoff,2-Solitoff,静止和运动呼吸子、瞬子、Compacton、Peakon等;考察了一些典型的非线性激发模式间如Compacton和Compacton,Peakon和Peakon,线孤子和y-周期孤子,Dromion和Dromion,Dromion和Solitoff之间等相互作用行为,发现其相互作用可以弹性的也可以是非弹性的,揭示了多样奇异而重要的非线性特性。
在2+1维分离变量解中引入适当的多值函数,构造出多种复杂的多值局域折叠孤立波形态,如蠕虫状折叠孤波、绳结型折叠孤波、帐篷式折叠孤波、蠕虫状Solitoff折叠孤波,蛹虫状dromion折叠孤波等.折叠孤波在所有方向都呈多值局域折叠形态,折叠子相互作用为弹性碰撞。分析了两类不同分离变量解基础上的局域激发模式的渐进行为,给出了折叠孤波渐进演化行为的一般特点以及折叠孤波相互作用为弹性的相移条件,并具体研究了折叠孤波的弹性相互作用行为(折叠子)。在2+1维分离变量解中引入多值函数和不同单值函数,得到了多种半折叠孤波激发,如钟状半折叠圈孤子、Compacton半折叠圈孤子、Peakon半折叠圈孤子等。半折叠孤子在一个方向上多值圈孤子折叠,而在另一方向上非线性单值局域。半折叠孤子之间的非线性相互作用是弹性,而半折叠孤子与单值局域孤子、半折叠孤子与折叠孤子以及更复杂的半折叠孤子、折叠孤子、单值局域相干孤子之间等多值和单值非线性激发模式间的相互作用是非弹性的,呈现出多样的非线性特性。这些研究展示了丰富多姿的多值局域折叠激发形态,为进一步了解和研究自然界的各种复杂非线性多值折叠现象奠定了基础。
第二部分,本文拓展和推广了新近发展的几种典型的直接代数方法和基于分离变量解的间接方法,获得多个2+1维非线性波动模型丰富的雅克比椭圆函数双周期波解和周期传播波模式,并得到长波极限情形下存在的非线性孤立波。
发展和应用Hirota双线性-θ函数方法,雅克比椭圆函数展开法,线性叠加法,F-函数展开法等分别求解2+1维2DsG方程,耦合ZK方程,2+1维KdV方程,2+1维长波短波共振相互作用方程,2+1维色散长波方程,获得丰富的雅克比椭圆函数双周期波解,描述了一些周期波形态及周期特性.长波极限下即椭圆函数模量m→1时,一些椭圆函数波退化得到相应的非线性孤立波解,如钟形孤波,单扭结孤波,马鞍型双扭结孤波,线孤子等。此外,还将椭圆函数展开方法拓展应用到离散差分非线性系统,获得了离散AL方程的周期波解和离散孤子,并揭示了亮、暗离散孤子的非线性特性。
从2+1维分离变量通解出发,引入适当的雅克比椭圆函数如雅克比正弦、余弦函数,雅克比第三椭圆函数及其倒数函数等,得到(2+1)维HBK方程和Maccari所得孤子方程的多种周期传播波,揭示了不含复变量和含复变量的两类非线性波动模型的周期传播波特性.这些椭圆函数波随着椭圆函数模量的变化呈现出不同的形态,可以是周期振荡波,也可以是Drmion格子,多Peakon激发形态等,长波极限下,可以退化为单值局域Dromions、多Lumps激发模式.椭圆函数传播波之间的相互作用是非弹性的,而在长波极限下相应相干孤子激发Dromion,Lumps之间的相互作用可以是弹性的,也可以是非弹性的。此外,我们还构造了在雅克比椭圆函数波背景下的Dromion孤子,这种新的复合波形态与海洋中发生的畸形极值波极其类似。
第三部分,本文从2+1维分离变量通解出发,提出了构造一种新非线性波即周期折叠波的简单方法,获得了的2+1维HBK方程和2+1维KdV方程的周期折叠波。我们基于2+1维非线性分离变量解,引入适当的多值函数和雅克比椭圆如雅克比椭圆正弦、余弦函数等,得到蠕虫状、帐篷状等多种复杂非线性周期折叠波。周期折叠波在一个方向上呈多值圈孤子折叠,而在另一方向上为单值周期振荡波,数学解析上可以视为特殊的半折叠波形态。周期折叠波的非线性相互作用是完全弹性的,在长波极限下退化为单折叠孤立波,相应折叠孤立波的非线性相互作用也是完全弹性的,除了相位变化,相互作用前后其形状和速度均不发生变化。这些研究有助于我们理解和认识自然界特殊的非线性周期折叠波现象。
Exploring nonlinear excitation patterns and complex nonlinear waves in high dimension nonlinear systems is an important topic in the research fields of nonlinear soliton theory,wave theory and nonlinear mechanics.There are various types of nonlinear folded phenomena and periodic waves in nature,which attract us to seek reasonable theoretical explanations in the study of nonlinear wave models.This paper focus on 2+1 dimensional nonlinear wave models (equations) originated from practical water and other nonlinear physical problems,seeking their exact solutions,revealing abundant nonlinear folded solitary wave and periodic propagating wave patterns and investigating their nonlinear interaction properties analytically and graphically.The main works and our achievements are outlined in three parts in the follows.
PartⅠ,the approaches multi-linear variable separation method,Painlevétruncated expansion method and generalized deformation mapping method that presented in recent years are extended to solve many 2+1 dimensional nonlinear physical models including 2+1 dimensional long wave short wave resonance interaction equation,Maccari system,generalized NNV equation,2+1 dimensional KdV equation,HBK equation and 2+1 dimensional variable coefficients BK equation,the corresponding exact variable separation solutions of them are obtained respectively.Based on 2+1 dimensional variable separation solutions,introducing proper single valued and multivalued functions,abundant nonlinear single valued and multivalued localized excitation patterns especially folded solitary wave excitations are derived.
Selecting proper single valued variable separation functions in the 2+1 variable separation solutions,we construct many types of single valued coherent soliton structures such as dromion,dromion lattice,solitoff,2-solitoff,static and kinetic breather,instanton,compacton, peakon,and so on.The interactions among these coherent localized excitations including the collisions of compaction and compacton,peakon and peakon,line soliton and y-periodic soliton,dromion and dromion,dromion and solitoff,et al,are investigated analytically and graphically,and found that can be elastic and nonelastic respectively,which reveal various exotic and important nonlinear characters and interaction properties.
Introducing proper multivalued functions in 2+1 dimensional variable separation solutions, we construct many types of mulitivalued localized solitary waves such as "worm" shape folded solitary wave,"rope knot" shape folded solitary wave,"tent" shape folded solitary wave,worm-solitoff folded solitary wave and worm-dromion folded solitary wave,et al. The folded solitary wave excitation is multivalued localized and folded in all directions and the interaction between foldons is elastic.We consider the asymptotic behaviors of the localized excitations produced on two families of variable separation solutions and give out the phase shifting conditions that lead to completely elastic interaction for the folded solitary waves,and we investigate the completely elastic interactions between two folded solitary waves(foldons) analytically and graphically.Introducing single valued and multivalued functions properly in the 2+1 dimensional variable separation solutions,we derive various semifolded solitary waves such as bell-like semifolded soliton,compaction-like semifolded soliton and peakon-like semifolded solitons,et al.The semifolded solitary wave excitations are multivalued loop soliton folded in one direction and single valued localized in other direction. The interactions between two semifolded solitons are elastic,while the interactions among other nonlinear multivalued and single valued excitation patterns such as the interactions between semifolded soliton and single valued localized soliton,semifolded soliton and folded soliton,and the interaction among semifolded soltion,folded soliton and single valued localized solitons,are nonelastic generally,which reveal different exotic and important nonlinear mechanical properties.Our research work expose various interesting and important localized multivalued folded excitations,which help us to further understand and explore complex nonlinear folded phenomena in nature.
PartⅡ,Applying the developed and extended direct algebraic methods that presented in recent years and based on the 2+1 dimensional variable separation solutions,abundant Jacobi elliptic function doubly periodic waves and periodic propagating wave patterns for many 2+1 dimensional nonlinear wave models are obtained,and in long wave limit solitary waves are derived as well.
We develop and apply the Hirota bilinear-θfunction method,Jacobi elliptic function expansion method,linear superposition method and F-expansion method respectively to solve many 2+1 dimensional nonlinear wave models including 2+1 dimensional 2DsG equation,the coupled ZK equation,2+1 dimensional KdV equation,2+1 dimensional long wave short wave resonance interaction equation and 2+1 dimensional dispersive long wave equation,abundant Jacobi elliptic function doubly periodic solutions are derived.These solutions show various periodic wave shapes and special periodic characters.In long wave limit,i.e,the modulus of the Jacobi elliptic function m→1,some elliptic function waves may degenerate into solitary wave solutions such as bell-like solitary wave,kink solitary wave,saddle-like solitary wave and line soliton as well.Further more,we also extend the Jacobi elliptic function expansion method to discrete differential-difference nonlinear systems.The periodic wave solutions and discrete solitons for discrete AL equation are obtained,and important nonlinear properties of discrete bright soliton and dark solitons are revealed.
Based on 2+1 dimensional variable separation solutions,introducing proper Jacobi elliptic functions such as sn,cn,dn and nd,et al.,we obtain many types of periodic propagating wave patterns for 2+1 dimensional HBK equation and the soliton equation presented by Maccari respectively,which denote the characters and nonlinear properties of periodic propagating waves for two classes of nonlinear wave models that possess complex variables or not.These elliptic function waves shows different forms as the modulus changes,which may be periodic oscillation waves,dromion lattic and multi-peakons,et al.In long wave limit, these elliptic function wave patterns may degenerate into single valued localized excitations dromion and multi-lumps.The interactions of elliptic function waves are nonelastic,while the interaction of the degenerated coherent soliton structures dromions and lumps may be elastic and nonelastic as well.Moreover,we also construct a new kind of combined wave possessing dromion solitons in the background of Jacobi elliptic periodic waves,which is very similar in the forms to the freak extreme waves that arise sometime in ocean.
PartⅢ,we present simple and effective technics to find new nonlinear waves the periodic folded waves based on 2+1 dimensional variable separation solutions,and the periodic folded waves of 2+1 dimensional HBK equation and 2+1 dimensional KdV equation are constructed respectively.Introducing suitable multivalued functions and Jacobi elliptic functions such as sn and cn appropriately,many types of complex nonlinear periodic folded waves such as wormlike periodic folded waves and tent-like periodic folded waves are derived.The periodic folded wave is multivalued loop soilton folded in one direction and single periodic oscillated in other direction,which may be view as a special type of semifolded wave in mathematical analytic. In long wave limit,the periodic folded waves may degenerate as single folded solitary waves. The interactions of the periodic folded waves and of their degenerated single solitary waves are completely elastic,i.e,there are no changes of the shapes and velocities before and after the interaction besides phases shifts.This work help us to further understand and acquaint such special periodic folded phenomena in nature.
引文
[1]R.Z.Sagdeev.Nonlinear Physics.London:Harword Academic Publisher,1988.
[2]D.Kaplan,L.Class.Understanding Nonlinear Dynamics.Springer-Verlag,1995.
[3]B.曼德布罗特.分形对象--形、机遇与维数.北京:世界图书出版公司,1999.
[4]李翊神,汪克林,郭光灿,汪秉宏.非线性科学选讲.合肥:中国科学技术出版社,1995.
[5]郝柏林.分岔、混沌、奇怪吸引子、湍流及其他.物理学进展,1993,3(1).
[6]陆同兴.非线性物理概论.合肥:中国科学技术出版社,2002.
[7]魏诺.非线性科学基础与应用.北京:科学出版社,2004.
[8]汪小帆,李翔,陈关荣.复杂网络理论及其应用.北京:清华大学出版社,2005.
[9]G.B.Whitham.Linear and Nonlinear waves.York:Wiley-Interscience Publication,1973.
[10]郭秉荣.线性与非线性导论.北京:气象出版社,1990.
[11]谷内俊弥,西原功休著,徐福元译.非线性波动.北京:原子能出版社,1981.
[12]V.I.Karpman.Nonlinear waves in dispersive media.Oxford Pergamon,1995.
[13]徐肇廷.海洋内波动力学.北京:科学出版社,1999.
[14]G.G.Stokes.On the theory of oscilatory waves.Camb.Trans.Phi.Soc.,8(1847)441.
[15]B.Remann.Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite.Gottingen Abhandlungen,8(1858)43.
[16]M.J.Abowitz,P.A.Clarkson.Soliton,nortlinear evolution equation and Inverse scattering.London:Cambridge University Press,1991.
[17]R.K.Dodd,J.C.Eilbeck,H.C.Morris.Soliton and Nonlinear Equation.Lodon:Academic Press,1984.
[18]E.Infeld,G.Rowlands.Nonlinear Waves,Soliton and Chaos.Cambridge,England:Cambridge University Press,2000.
[19]C.H.Gu,B.L.Guo,Y.S.Li,C.W.Cao,C.Tian,G.Z.Tu,M.L.Ge.Soliton Theory and Its Applications.New York:Springer-Verlag,1995.
[20]谷超豪,胡和生,周子翔.孤子理论中的达布变换及其几何应用.上海:上海科学技术出版社,2001
[21]C.Rogers,W.K.Schief.Backlund and Darboux Transformations gemetry and modern applications in soliton theory.Cambridge University Press,Cambridge Texts in Applied Mathematics,2002.
[22]谷超豪等.孤立子理论与应用.杭州:浙江科学技术出版社,2002.
[23]李翊神编著.孤子与可积系统.上海:上海科技教育出版社,1997.
[24]陈陆君,梁昌洪.孤立子理论及其应用.西安:西安电子科技大学出版社,1997.
[25]F.Abdullaev,S.Darmanyan,P.Khabibullaev.Optical Solitons.Springer-Verlag,1993.
[26]倪皖荪,魏荣爵.水槽中的孤波.上海:上海科技教育出版社,1994.
[27]刘式达,刘式适.孤波与湍流.上海:上海科技教育出版社,1994.
[28]楼森岳,唐晓艳.非线性数学物理方法.北京:科学出版社,2006.
[29]闫振亚.复杂非线性波的构造性理论及其应用.北京:可许技术出版社,2007.
[30]郭柏灵.非线性演化方程.上海:上海科技教育出版社,1995.
[31]刘式达,刘式适.物理学中的非线性方程.北京:北京大学出版社,2000.
[32]D.J.Korteweg and G.de Vries.On the change of form of long waves adavancing in a rectangular channel,and on a new type of long stationary waves.Phil.Mag.39(1895)422.
[33]L.Dolan.Gauge symmetry in background charge conformal field theory Nucl.Phys.B 489(1997)245;
N.Flytzanis,S.Pnevmatikos and M.Remoissenet.Kink,breather and asymmetric envelope or dark solitons in nonlinear chains.I.Monatomic chain.J.Phys.C:Solid State Phys.18(1985)4603.
[34]D.H.Peregrine.Water waves,nonlinear Schrodinger equations and their solutions.J.Austral.Math.Soc.Ser.B,25(1983)16.
[35]A.C.Scott,F.Y.F.Chu and D.W.Mclanghlin.The soliton A new concept in applied science.Proc.IEEE,16(1973)1443.
[36]M.J.Ablowitz and J.F.Ladik.Nonlinear differential-difference equations.J.Math.Phys:16(1975)598.
[37]D.N.Christodoulides,R.I.Joseph.Discrete self-focusing in nonlinear arrays of coupled waveguides.Opt.Lett.13(1988)794;
A.Ttrombettoni,A.Smerzi.Discrete solitons and breathers with dilute Bose-Einstein condensates.Phys.Rev.Lett.86(2001)2353;
O.O.Vakhnenko.Solitons on a zigzg-runged Ladder Lattice.Phys.Rev.E 64(2001)067601.
[38]B.B.Kadomtsev and V.I.Petviashavili.On the stability of solitary waves in weakly dispersing media.Sov.Phys.Dokl.,15(1970)539;
M.J.Ablowitz and H.Segur.On the evolution of packets of water waves.J.Fluid.Mech.,92(1979)691.
[39]M.J.Ablowitz,H.Segur.Solitons and Inverse Scattering Transform.SIAM,Philadelphia,PA,1981.
[40]A.Davey and K.Stewartson.On three-dimensional packets of surface waves.Proc.R.Soc.A,338(1974)101.
[41]L.P.Nizhnik.Integration of multidimensional nonliner equations by the method of the inverse problem.Sov.Phys.Dolk.,25(1980)706;
A.P.Veselov and S.P.Novikov.Finitezone,two-dimensional,potential Schodinger opters,explicit formulas and evolution equations.Sov.Math.Dolk.,30(1984)588;
S.P.Novikov and A.P.veselov.Two-dimensional Schr?dinger operator:Inverse scattering transform and evolutional equations.Physica D,18(1986)267.
[42]S.Y.Lou,X.B Hu.Infinitely many Lax pairs and symmetry constraints of the KP equation.J.Math.Phys,1997;38:6401.
[43]P.Boiti,J.J.P.Leon,M.Manna,F.Pempinelli.Inverse Problems 3(1987)1205;
G.Paquin and P.Winternitz.Group theoretical analysis of dispersive long wave equations in two space dimensions Physica D 46(1990)122.
[44]J.S.Russell.Report of the committee on waves.Rep.Meet,Brit.Assoc.Adv.Sci.7th Livepoll (1837)417,London,John Murray.
[45]G.B.Airy.Tides and waves.Encyclopedia Metropolotana,5(1845)241.
[46]M.J.Boussinesq.Théorie de I'intumescence appelee onde solitarire ou de translationse propageant dans un canal recangulaire.C.R.Acad,Sc.Paries,72(1871)755.
[47]L.Rayleigh.On waves.Philos,Mag.,1(1976)257.
[48]A,Fermi,J.Pasta,S.Ulam.Studies of nonlinear problems.Los Almos Report.,1955,LA1940.
[49]J.K Perring,T.H.R.Skyrme.A model unified field equation.Nucl.Phys.,31(1962)550.
[50]A.Seeger,H.Donth,A.Kochendorfer.Theore der versetzungen in eindimensionalen,atomreihen:Ⅲ.Versetzungen,eigenbewegtmgen und ihre wecselwirkung.Z.Phys.,134(1953)173.
[51]N.J.Zabusky,M.D.Kruslad.Interaction of solitons in a collisionless plasma and therecurrence of initial states.Phys.Rev.Lett.,15(1965)240.
[52]M.Tode.Wave propagation in anharmonic lattices.J.Phys.Soc.Jpn.,23b(1967)501.
[53]R.Camassa and D.D.Holm,An integrable shallow water equation with peaked solitons.Phys.Rev.Lett.,71(1993)1661.
[54]P.Rosebau and J.Hyman.Compactons-Solitons with finite wave length.Phys.Rev.Lett.,70(1993)564.
[55]K.Konno,Y.Ichikawa and M.Wadati.A loop soliton propagating along a stretched rope.J.Phys.Soc.Jpn.,50(1981)1025.
[56]M.Boiti,J.J.Leon,L.Martina and F.Pempinelli.Scattering of localized solitons in the plane.Phys.Lett.A,132(198)432.
[57]A.Hasegawa,F.Tappert.Transmission of stationary nonlinear optical pulses in despersive dieledtric fibers.I.anomalous dispersion.Appl.Phys.Lett.,23(3)(1973)142;
A.Hasegawa,F.Tappert.Transmission of stationary nonlinear optical pulses in despersive dieledtric fibers.Ⅱ.normal dispersion.Appl.Phys.Lett.,23(4)(1973)171.
[58]L.F.Mollenauer,R.H.Stolen,J.P.Gordon.Expermental observation of picosecond pulse narrowing and soliton in optical fibers.Phys.Rev.Lett.,45(1980)1095.
[59]P,Emplit,J.P.Hamaide,F.Reynaud,C.Froehly and A.Barthelemy.Picosecond steps and dark pulses through nonlinear single mode fibers.Opt.Commun.,62(1987)374.
[60]M.C.Ferriera,R.A.Kraenkel and A.Zenchuk.Soliton-Cuspon interaction for the Cammassa-Holm equation.J.Phys.A:Math.Gen.32(1999)8665.
[61]R.Beals,D.H.Sattinger and J.S.Zmigielski.Peakon-Antipeakon interaction.J.Nonlinear Math.Phys.,8(2001)1347.
[62]T.F.Qian and M.Y.Tang,Peakons and periodic cus waves in a generalized Camassa-Holm equation.Chaos,Soliton and Fractals,12(2001)1347.
[63]V.I.Arnold.Mathematical method of clasical Mechanics.Berlin:Spring-Verlag,1978.
[64]V.O.Vakhnenko.Solitons in a nonlinear model medium.J.Phys.A:Math.Gen.25(1992)4181;V.O.Vakhnenko and E.J.Parkes.The two loop soliton solution of the Vakhnenko equation.Nonlinearity,11(1998)1457.
[65]S.Matsutani.The relation of lemniscate and a loop soliton as 3/2 and 1 spin fields along the modified korteweg-de vries equation.Mod.Phys,Lett.A,10(1995)717.
[66]M.Schleif,R.Wunsch and T.Meissner.Self-Consistent Pushing and Cranking Corrections to the Meson Fields of the Chiral Quark-Loop Soliton.Int.J.Mod.Phys.E,7(1998)121;
M.Schleif and R.Wunsch.Eur.Phys.A,1(1998)171.
[67]H.Kahuhata and K.Knono.Loop Soliton Solutions of String Interacting with External Field.J.Phys.Soc.Jpn.68(1999)757.
[68]王明亮.非线性发展方程与孤立子.兰州:兰州大学出版社,1990.
[69]S.Y.Lou.Abundant symmetries for the 1+1 dimensional classical Liouville field theory.J.Math.Phys.35(1994)2336;
S.Y.Lou.Syrnmetries of the KdV equation and four hierarchies of the integrodifferential KdV equations.J.Math.Phys.35(1994)2390.
[70]M.J.Ablowitz,A.Ramani,and H.Segur.A connection between nonlinear evolution equations and ordinary differential equations of P-type.J.Math.Phys.,21(2980)715;1006.
[71]M.Jimbo,et al.Painlevé test for theself-dual Yang-Milis equation.Phys.Lett.A,92(1982)59;
A.P Fordy,A.Pickering.Analysing negative resonances in the Painlevétest.Phys.Lett.A,160(1991)347.
[72]F.Calogero,W.Eckhaus.Nonlinear evolution equations,rescalings,model PDES and their integrability:I.Inverse Problem,3(1987)229;
F.Calogero,X.D.Ji.J.Math.Phys.32(1991)2703.
[73]S.Y.Lou,X.Y.Tang,Q.P.Liu,and T.Fukuyama.Second Order Lax Pairs of Nonlinear Partial Differential Equations with Schwarzian Forms.Z.Naturforsch A,57(2002)737;
X.Y.Tang,H.C.Hu.Characteristic Manifold and Painlev é Integrability:Fifth-Order Schwarzian Korteweg-de Type Equation.Chin.Phys.Lett.,19(2002)1225.
[74]C.S.Garner,J.M.Greene,M.D.Kruskal and R.M.Miura,Method for solving Korteweg de Vries equation,Phys.Rev.Lett.,19(1967)1095.
[75]H.D.Wahlquist,F.B.Estabrook.Backlund transformation for soliton of the KdV equation.Phys.Rev.Lett.,31(1973)1386.
[76]G.Darboux.Sur une peoposition relative auxéquations linéaires.C.R.Acad,Sci.Paris,94(1882)1456.
[77]R.Hirota.Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons.Phys.Rev.Lett.,27(1971)1192.
[78]R.Hirota.The Detect Method in Soliton Theory.Cambridge Unversity Press,2004.
[79]J.Matsukidaira,J.Satsuma.Integrable four-dimensional nonlinear Lattice expressed by trinear form.J.Phys.Soc.Jpn.,59(1990)3413.
[80]G.W.Bluman,S.Kumei.Symmetries and Differential Equation.Appl.Math.Sci.81,Springer,Berlin,1989.
[81]P.J.Olver.Application of Lie Groups to Differential Equation,2nd ed.Graduate Texts Math,107,New York:Springer,1993.
[82]P.A.Clarkson,M.D.Kruskal.New similarity solutions of the Boussinesq equation.J.Math.Phys.,30(1989)2201.
[83]S.Y.Lou.A note on the new similarity reductions of the Boussinesq equation.Phys.Lett.A,151(1990)133.
[84]J.Weiss,M.Tabor,G.Garnevale.The Pailevéproperty for partial differential equations.J.Math.Phys.,24(1983)522.
[85]S.Y.Lou.Some special types of Multisoliton solutions of the Nizhnik-Novikov-Vesselov equation.Chin.Phys.Lett.,17(2000)781;
J.Lin,S.Y.Lou,K.L Wang.Multisoliton solutions of the Konopelchenko-Dubrovsky equation.Chin.Phys.Lett.,18(2001)1173;
W.H.Huang,D.J.Yu,M.Z.Jin.Some special types of Multisoliton solutions of the Modified Kapdomtshev-Petviashvilli equation.Commun.Theor.Phys.,40(2003)262.
[86]M.L.Wang,Solitary wave solutions for variant Boussinesq equations.Phy.Lett.A,199(1995)169.
M.L.Wang,et al.Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics.Phys.Lett.A,216(1996)67.
[87]Y.T.Gao,B.Tian.New families of exact solutions to the intergrable dispersive long wave equations in(2+l)-dimensional spaces.J.Phys.A,29(1996)2895;
张解放.长水波近似方程的多孤子解.物理学报,47(1998)1416.
[88]P.W.Dolye.Separation of variables for scalar evolution equations in one space dimension.J.Phys.A:Math.Gen.,29(1996)7581.
[89]E.D.Belokolos,et al.Algebro-geometric approach to nonlinear integrable equations.Berlin:Springer-Verlag,1994.
[90]S.Y.Lou and J.Z.Lu.Special solutions from variable separation approach:Davey-Stewartson equation.J.Phys.A:Math.Gen.29(1996)4209.
[91]S.Y.Lou.On the coherent structures of Nizhnik-Novikov-Veselov.Phys.Lett.A,277(2000)94.
[92]S.Y.Lou and H.Y.Ruan.Revisiatio of the localized excitations of the(2+1)-dimensional KdV equation.J.Phys.A:Math.Gen.,34(2001)305.
[93]X.Y.Tang,S.Y.Lou and Y.Zhang.Localized exciatations in(2+1)-dimensional system.Phys.Rev.E,66(2002)046601.
[94]X.Y.Tang and S.Y.Lou.Extended Multilinear variable separation approach and multivalued localized excitations for some(2+1)-dimensional integrable systems.J.Math.Phys.44(2003)4000.
[95]徐桂琼,李志斌.构造非线性发展方程孤波解的混合指数方法.物理学报,51(2002)946;
扩展的混合指数方法及其应用,51(2002)1424.
[96]L.Huibin and W.Kelin.Exact solutions for some coupled nonlinear equations.J.Phys.A:Math.Gen.,23(1990)4097;E.J.Parkes.Comput.Phys.Commun.,98(1996)288;
H.M.Li,F.M.Wu.Soliton solutions of Bose-Einstein Condensate in Linear Magnetic field and Time-Dependent Laser Field.Chin.Phys.Lett.,21(2004)1425.
[97]E.G.Fan.Extended tanh-function method and its applications to nonlinear equations.Phys.Lett.A 277(2000)212;
F.D.Xie,Y.Zhang,Z.S.Lü.Symbolic computation in nonlinear evolution equation:application to(3+1)-dimensional Kadomtsev-Petviashvili equation.Chaos,Solitons and Fractals,24(2005)257;
[98]C.T.Yan.A simple transformation for nonlinear waves.Phys.Lett.A,224(1996)77;
闫振亚,张鸿庆,范恩贵.一类非线性演化方程新的显式行波解.物理学报,48(1999)1;
贺峰,郭启波,刘辽.用三角函数法获得非线性Boussineeq方程的广义孤子解.物理学报,56(2007)4326.
[99]郑埘,张鸿庆.一个非线性方程的显式行波解.物理学报,49(2000)389;
张桂戌,李志埘,段一士.非线性波方程的精确孤立波.中国科学,30(2000)1103;
吕克璞,石玉仁,段文山.KdV-Burgers方程的孤波解.物理学报,50(2001)2074;
黄文华,张解放,盛正卯,Boussinesq方程的新显式精确行波解.浙江大学学报(理学版),30(2003)145;
黄定江,张鸿庆.扩展的双曲函数法和Zakharov 方程组的精确孤波解.物理学报,53(2004)2434.
[100]Z.Y.Yan.Generalized method and its application in the higher-order nonlinear Schordinger equation in nonlinear optical fibres.Chaos.Soliton and Fractals,16(2003)759;
B.Li and Y.Chen.Nonlinear partial differential equations solved by projective Riccati equations ansatz.Z.Naturforsch,58a(2003)511.
[101]雍雪林,张鸿庆.推广的投影Riccati方程法及其应用.物理学报,54(2005)2514;
C.Q.Dai,J.M.Zhu,J.F.Zhang.New exact solutions to the mKdV equation with variable coefficients.Chaos,Solitons and Fractals,27(2006)881.
[102]S.K.Liu,Z.T.Fu,S.D.Liu.Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations.Phys.Lett.A,289(2001)69;
Z.T.Fu,S.K.Liu,S.D.Liu and Q.Zhao.New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations,Phys.Lett.A,290(2001)72.
[103]刘式达,付遵涛,刘式适,赵强.Jacobi椭圆函数展开法及其在求解非线性波动方程中的应用.物理学报,50(2001)2068;
刘式适,付遵涛,刘式达,赵强.一类非线性方程的新周期解.物理学报,51(2002)10;
刘式适,付遵涛,刘式达,赵强.一类非线性方程的新周期解.物理学报,51(2002)10;
刘式达.付遵涛,刘式适,赵强.非线性波动方程的Jacobi椭圆函数包络周期解.物理学报,51(2002)718.
[104]E.G.Fan and J.Zhang.Applications of the Jacobi elliptic function method to special type nonlinear equations.Phys.Lett.A,305(2002)383;
E.G.Fan and B.Y.C.Hon.Doubly periodic solutions with Jacobi ellptic functions for two generalized Hirota-Satsuma coupled KdV equation.Phys.Lett.A,292(2002)335.
[105]Z.Y.Yan.Extended Jacobi elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations.Commput.Phys.Commun.,148(2002)30;
Y.Z.Yan.The extended Jacobi elliptic function expansion method and its application in the generalized Hirota-Samuma coupled KdV equation.Chaos,Soliton and Fractals,15(2003)575;
Z.Y.Yan.Abundant families of Jacobi elliptic function solutions of the(2+1)-dimensional intergrable Davey-Stewartson type equation via a new method.Chaos,Soliton and Fractals,18(2003)299.
[106]Z.Y.Yan.The new extended Jacobian elliptic function expansion algorithm and its applications in nonlinear mathematical physics equations.Compt.Phys.Commun.,153(2003)145.
[107]J.M.Zhu,et al.General Jacobian elliptic function expansion method and its applications.Chin.Phys.,13(2004)0798;
X.Q.Zhao,H.Y.Zhi,H.Q.Zhang.Improved Jacobi-function method with symbolic computation to construct new doubly -periodic solution forthe generalized Ito system.Chaos,Soliton and Fractals,28(2006)112;
吴国将,等.扩展的Jacobi椭圆函数展开法和zakharov方程组的新的精确周期解.物理学报,56(2007)5054.
[108]D.Z.Lü.Jacobi elliptic function solutions for two variant Boussinesq equations.Chaos,Soliton and Fractals,24(2005)1737.
[109]Z.T.Fu,S.D.Liu,S.K.Liu,Q.Zhao.New exact solutions to KdV equations with variable cofficients or forcing.Applied Mathematics and Mechanics,25(2004)73.
[110]X.J.Lai and J.F.Zhang.Exact localized and periodic solutions of the Ablowitz-Ladik Discrete nonlinear Schrodinger system.Z.Naturforsch,60a(2005)573;
C.Q.Dai,J.P.Meng and J.F.Zhang.Symbolic computation of extended Jacobian elliptic function algorithm for nonlinear differential-differen equation.Commun.Theor.Phys.,43(2005)471;
C.Q.Dai,J.F.zhang.Jacobian elliptic function method for nonlinear differential difference equation.Chaos,Solitons and Fractals,27(2006)1042.
[111]Y.B.Zhou,M.L.Waang,Y.M.Wang.Periodic wave solutions to a coupled KdV equations with variable coefficients.Phys Lett A,308(2003)31;
J.L.Zhang,M.L.Wang,D.M.Cheng and Z.D.Fang.The Periodic Wave Solutions for Two Nonlinear Evolution Equations.Commun.Theor.Phys.,40(2003)129.
[112]D.S.Wang,H.Q.Zhang.Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation.Chaos,Solitons and Fractals,25(2005)601;
B.A.Li,M.L.Wang.Applications of F-expansion method to the coupled KdV equation.Chin.Phys.,14(2005)1698.
[113]D.S.Wang,Y.F.Liu,H.Q.Zhang.Symbolic computation and families of Jacobi elliptic function solutions of the(2+1)-dimensional Nizhnik-Novikov-Veselov equation.Applied Mathematics and Computation.,168(2005)823;
Emmanuel Yomba.The extended F-expansion method and its application for solving the nonlinear wave,CKGZ,GDS,DS and GZ equations.Phys.Lett.A,340(2005)149.
[114]M.I.Wang,X.Z.Li.Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations.Phys.Lett.A,343(2005)48.
[115]J.Chen,H.S.He,K.Q.Yang.A Generalized F-expansion Method and Its Application in Highdimensional Nonlinear Evolution Equation.Commun.Theor.Phys.,44(2005)307;
Y.J.Ren,H.Q.Zhang.A generalized F-expansion method to find abundant families of Jacobi Elliptic Func- tion solutions of the(2+1)-dimensional Nizhnik-Novikov-Veselov equation.Chaos,Solitons and Fractals,27(2006)959.
[116]J.F.Zhang,C.Q.Dai,Q.Yang,J.M.Zhu.Variable-coefficient F-expansion method and its application to nonlinear Schrodinger equation.Optics Communications,252(2005)408.
[117]Z.Y.Yan.Jacobi elliptic function solutios of nonlinear wave equations via the new sinh-Gordon equation expansion method.J.Phys.A,36(2003)1961;
Z.Y.Yan.Elliptic function solutions of(2+1)-dimensional long wave-short wave resonance interaction equation via a shinh-Gordon expansion method.Z.Naturforsch.59a(2004)23.
[118]Z.Y.Yan.New weierstrass semi-rational expansion method to doubly periodic solutions of soliton equation.Commun.Theor.Phys.,34(2005)391;
Z.Y.Yan.New doubly periodic solutions of nonlinear evolution equations via Weierstrass elliptic function expansion algorithm.Commun.Theor.Phys.,42(2004)645;
李德生,张鸿庆,构造孤子方程德Weierstrass椭圆方程函数解的一个新方法.物理学报,54(2005)5540.
[119]K.W.Chow,D.W.C.Lai.Coalescence of wavenumbers and exact solutions for a system of coupled Schordinger equations.J.Phys.Soc.Jpn.,67(1998)3721.
[120]D.W.C.Lai,K.W.Chow.Positon and Dromion solutions of the(2+1)dimensional long wave short wave resonance interaction.J.Phys.Soc.Jpn.,68(1999)1847.
[121]D.W.C.Lai,K.W.Chow.Coalescence of Ripplons,Breathers,Dromions and Dark Solitons.J.Phys.Soc.Jpn.,70(2001)666.
[122]J.H.He,X.H.Wu.Exp-function method for nonlinear wave equations.Chaos,Soliton and Fractals 30(2006)700;
S.Zhang.Application of Exp-function method to a KdV equation with variable coefficients.Phys.Lett.A,365(2007)448;
S.D.Zhu.Exp-function Method for the Hybrid-Lattice System.International Journal of Nonlinear Sciences and Numerical Simulation,8(2007)461.
[123]S.Y.Lou and G.J.Ni.The relations among a special type of solutions in some(D+1)dimensional nonlinear equations.J.Math.Phys.,30(1989)1614;
E.G.Fan.Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematica Physics.Chaos,Solitons and Fractals,16(2003)819.
[124]H.M.Li.Mapping deformation method and its application to nonlinear equations.Chin.Phys.,11(2002)1111;
李画眉,(3+1)维Nizhnik-Novikov-Veselov方程的孤子解和周期解.物理学报,51(2002)467;
W.H.Huang and M.Z.Jin.Explicit and exact travelling plane wave solutions of the (2+1)-dimensional Boussinesq equation.Chin.Phys.12(2003)361;
C.L.Zheng and L.Q.Chen.A general mapping approach and new travelling wave solutions to(2+1)dimensional Boussinesq equation.Commun.Theor.Phys.,41(2004)671.
[125]M.Boiti,J.P.Leon,and F.Pempinelli.Multidimensional solitons and their spectral transforms.J.Math.Phys.,31(1990)2612.
[126]A.S.Fokas and P.M.Santini.Dromions and a boundary value problem for the Davey-Stewartson 1 equation.Physica D,44(1990)99.
[127]J.Hietarinta.One-dromion solutions for genetic classes of equations.Phys.Lett.A,1990,149113;
S.Y.Lou.Generalized dromion solutions of the(2+1)-dimensional KdV equation J.Phys.A:Math.Gen,28(1995)7227.
[128]R.Radha and M.Lakshmanan.Dromion like structures in the(2+1)-dimensional breaking soliton equation Phys.Lett.A,197(1995)7;
R.Radha and M.Lakshmanan.Generalized dromions in the(2+1)-dimensional long despersive wave(2LDW) and scalar nonlinear schrodinger(NLS)equations.Chaos,Soliton and Fractals,10(1999)1821.
[129]阮航宇.(2+1)维NLSQ方程和KP方程的多Dromion结构和相互作用的研究.物理学报,48(1999)1781;
阮航宇.可积模型中孤子相互作用的研究物理学报,50(2001)369
[130]S.Y.Lou.Dromion,Dromions lattice,breathers andinstantons of the Davey-Stewartson equation.Phys.Scripta.65(2002)2.
[131]X.Y.Tang and S.Y.Lou.Exact solutions of(2+1)-dimensional dispersive long wave equation.Phys.Rev.E.66(2000)036605;
X.Y.Tang and S.Y.Lou.Abundant coherent structures of the dispersive long wave equation in(2+1)-dimensional spaces.Chaos,Soliton and Fractals,14(2002)1451.
[132]X.Y.Tang,C.L.Chen and S.Y.Lou.Localized solutions with chaotic and fractals behaviour in (2+1)-dimensional dispersive long wave system.J.Phys.A:Math.Gen.,35(2002)L293.
[133]S.Y.Lou.(2+1)-dimensional compacton solutions with and without completely elastic interaction perties.J.Phys.A:Math.Gen.35(2002)10691.
[134]J.Lin and H.M.Li.Painleve integrablity and abundant localized structures of(2+1)-dimensional Higher-Order Broer-Kaup system.Z.Naturforsch.,57a(2002)929.
[135]J.F.Zhang,W.H.Huang and C.L.Zheng.Exotic localized coherent structures of new(2+1)-dimensional Soliton equation.Commun.Theor.Phys.,38(2002)517.
[136]张解放,黄文华,郑春龙.一个新(2+1)维非线性演化方程的相干孤子结构.物理学报,51(2002)2676.
[137]X.Y.Tang and S.Y.Lou.A variable separation approach for integrable and nonlintegrable models:coherent structures of 2+1 dimensional KdV equation.Commun.Theor.Phys.,38(2002)1;
J.Lin,F.M.Wu.Fission and fusion of localized coherent structures for a(2+1)-dimensional KdV equation.Chaos,Solitons and Fractals,19(2004)189.
[138]W.H.Huang,J.F.Zhang and Z.M.Sheng.Coherent soliton structures of the(2+1)-dimensional long wave short wave interaction equation.Chin.Phys.,11(2002)1101.
[139]J.F.Zhang,J.P.Meng,and W.H.Huang.A New Class of(2+1)-Dimensional Localized Coherent Structures with Completely Elastic and Non-elastic Interactive Properties.Commun.Theor.Phys.,42(2004)161.
[140]C.L.Zheng,J.F.Zhang,W.H.Huang,L.Q.Chen.Peakon and Foldon excitations in a(2+1)-dimemsional breaking soliton system.Chin.Phys.Lett.,20(2003)783.
[141]J.F.Zhang,J.P.Meng,C.L.Zheng,W.H.Huang.Folded solitary waves and foldons in the(2+1)- dimensional breaking soliton equation.Chaos,Solitons and Fractals,20(2004)523.
[142]K.Z.Hong,B.Wu,and X.F.Chen.Painlev éAnalysis and Some Solutions of(2+1)-Dimensional:Generalized Burgers Equations.Commun.Theor.Phys.,39(2003)393;
徐昌智,张解放.(2+1)维非线性Burgers方程变量分离解和新型孤波结构.物理学报,53(2004)2407.
[143]阮航宇,陈一新.(2+1)维非线性薛定谔方程的环孤子,dromion,呼吸子和瞬子.物理学报,50(2001)586.
[144]S.Y.Lou.Localized excitations of the(2+1)-dimensional sine-Gordon system.J.Phys.A:Math.Gen.36(2003)3877.
[145]S.Y.Lou,J.Lin and X.Y.Tang.Painlevé integrabity and multi-dromion solutions of the(2+1)dimensional AKNS system.Eur.Phys.J.B22(2001)473.
[146]S.Y.Lou,X.Y.Tang and C.L.Chen.Fractal solutions of the Nizhnik-Novikov-Veselov equation.Chin.Phys.Lett.,19(2002)769;
C.L.Zheng,J.F.Zhang.General solution and fractal localized structures for the(2+1)-demensional generalized Ablowitz-Kaup-Newell-Segur system.Chin.Phys.Lett.,19(2002)1399;
C.L.Zheng.Loc alized coherent structures with chaotic and fractal behaviors in a modified(2+1)-dimensional long dispersive wave system.Commun.Theor.Phys.40(2003)25.
[147]郑春龙,(2+1)维非线性系统的局域激发模式及其分形和混沌行为研究.上海大学博士学位论文,上海,2005.
[148]X.Y.Tang and S.Y.Lou.Folded Solitary waves and foldons in(2+1)dimensions.Commun.Theor.Phys.40(2003)62.
[149]W.H.Huang,J.F.Zhang.Folded localized excitation of the Maccari system.Acta Physica Polonica B,35(2004)2051;
W.H.Huang,J.F.Zhang,W.G.Qiu.Compacton and foldon excitations in the generalized(2+1)-dimensional Nizhnik-Novikov-Veselov equation[J].Z.Naturforsch,59a(2004)250;10.
J.F.Zhang,W.H.Huang.Folded localized excitations of the(2+1)-dimensional Broer-Kaup equation.Commun.Theor.Phys.,40(2003)533.
C.L.Zheng,L.Q.Chen,J.F.Zhang.Muliti-valued solitary waves in mulitidimensional soliton system.Chin.Phys.,13(2004)592;
J.F.Zhang,Z.M.Lu and Y.L.Liu.Folded solitary waves and foldons in the(2+1)-dimensional long dispersive wave equation.Z.Naturforsch,58a(2003)280.
[150]S.Y.Lou,J.Yu,X.Y.Tang.Higher Dimensional Integrable Models from Lower Ones via Miura Type Deformation Relation.Z.Naturforsch,55a(2000)867;
J.Lin,S.Y.Lou,k.I.Wang.Highdimensional Virasoro integrable models and exact solutions.Phys.Lett.A,287(2001)257;
X.Y.Tang,Z.F.Liang.Variable separation solutions for the (3+1)-dimensional Jimbo-Miwa equation.Phys.Lett.A,351(2006)398.
[151]J.F.Zhang.Exotic localized coherent structures of the(2+1)-dimensional dispersive long wave equation.Commun.Theor.Phys.,37(2002)277.
J.F.Zhang.Homogenous balance method and chaotic and fractal solutions for the NNV equation.Phys.Lett.A,313(2003)401.
[152]C.L.Zheng and L.Q.Chen.Semifolded localized coherent structures in generalized(2+1)-dimensional Korteweg de Vries system.J.Phys.Soc.Jpn.,73(2004)293.
J.P.Fang,C.L.Zheng and L.Q.Cheng.Semifolded localized structures in three demensional soliton system.Commun.Theor.Phys.,42(2004)175.
[153]C.L.Bai,C.J.Bai,H.Zhao.Combined structures of single valued and multiple valued localized excitations in higher-dimensional soliton system.Phys.Lett.A,Phys.Lett.A,357(2006);
C.L.Bai,H.Zhao.New types of interactions between solitary waves in(2+1)-dimensions.Chaos,Solitons and Fractals,32(2007)375.
[154]C.Q.Dai,G.Q.Zhou,J.F.Zhang.Exotic localized structures based on variable separation solution of(2+1)-dimensional KdV equation via the extended tanh-function method.Chaos,Solitons and Fractals,33(2007)1458;
C.Q.Dai,J.F.Zhang.Variable separation solutions for the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov equation.Chaos,Solitons and Fractals,33(2007)564.
何宝钢,徐昌智,张解放.物理学报,54(2005)5525
[155]张解放.非线性波动模型的相干结构及其相互作用.上海大学博士论文,2003.
[156]来娴静.孤子系统的多线性变量分离及其局域激发研究.浙江师范大学硕士论文,2005.
[157]阮航宇,高维模型的可积性和孤子相互作用性质的研究.博士论文,浙江大学,2000.
[158]J.Lin,X.M.Qian.The interaction of localized coherent for a(2+1)-dimensional system.Phys.Lett.A,313(2003)93;
H.Y.Ruan.The interaction of Solitons in(2+1)-dimensional Modified Nizhnik-Novikov-Vesselov equation.J.Phys.Soc.Jpn.,73(2004)566.
[159]C.L.Bai,C.J.Bai,H.Zhao.The interactions of localized coherent structures for a(2+1)-dimensional system.Chaos,Solitons and Fractals,34(2007)919.
[160]张解放,韩平(2+1)维Broer-Kaup方程的局域相干结构.物理学报,51(2002)705;
张解放,刘宇陆.高阶(2+1)维Broer-Kaup方程的局域相干结构,应用数学和力学,23(2002)489.
[161]C.l.Zheng,J.P.Fang and L.Q.Chen.New variable separation excitations of a(2+1)-dimensional Broer-Kaup-Kupershmidt system via an extended mapping approach.Z.Naturforsch,59a(2004)912;
C.L.Zheng,J.P.Fang.L.Q.Chen.Localized excitations with and without propagating properties in(2+1)-dimensions obtained by a mapping approach.Chin.Phys.,14(2005)0676;
C.L.Zheng,G.P.Cai,J.Y.Qiang,Chaos,solitons and fractals in (2+1)-dimensional KdV system derived from a periodic wave solution.Chaos,Solitons and Fractals,34(2007)1575.
[162]C.L.Zheng,J.P.Fang.L.Q.Chen.New variable separation excitations of(2+1)-dimensional dispersive long-water wave system obtained by an extended mapping approach.Chaos,Solitons and Fractals 23(2005)1741;
C.L.Zheng,J.P.Fang.L.Q.Chen.Localized excitations in(2+1)-dimensions obtained by a mapping approach.Chin.J.Phys.,43(2005)17.
[163]J.P.Fang,C.L.Zheng.New exact excitations and soliton fssion and fusion for the(2+1)-dimensional Broer-Kaup-Kupershmidt system.Chin.Phys.,14(2005)0669.
[164]C.L.Zheng,H.P.Zhu,L.Q.Chen.Exact solution and semifolded structures of generalized Broer- Kaup system in(2+1)-dimensions.Chaos,Solitons and Fractals,28(2005)187;
扩展的形变映射方法和(2+1)维破裂孤子方程的新解.物理学报.55(2006)511.
[165]Z.P.Yan.A class of doubly periodic wave solutions for the generalized Nizhnik-Novikov-Veselov equation.Phys.Lett.A,337(2005)55.
[166]Y.Emmanuel,Z.P.Yan.Fission,fusion and annihilation in the interaction of localized structures for the(2+1)-dimensional generalized Broer-Kaup system.Chaos,Solitons and Fractals,28(2006)650.
[167]W.H.Huang,Y.L.Liu,Z.M.Lu.Doubly periodic wave and folded solitary wave solutions for(2+1)-dimensional higher-order Broer-Kaup equation.Chaos,Solitons and Fractals,31(2007)54.
[168]R.M.Wang,J.Y.Ge,C.Q.Dai and J.F.Zhang.Construction of New Variable Separation Excitations via Extended Projective Ricatti Equation Expansion Method in(2+1)-Dimensional Dispersive Long Wave Systems.Int.J.Theor.Phys.,46(2007)105.
[169]Z.Y.Ma,Y.H.Hu.Solitons,chaos and fractals in the(2+1)-dimensional dispersive long wave equation.Chaos,Solitons and Fractals,34(2007)1667.
[170]P.A.Lindgard,H.Bohr.Magic Numbers in Protein Structures.Phys.Rev.Lett.77(1996)779;
S.W.Lockless,R.Ranganathan.Evolutionarily Conserved Pathways of Energetic Connectivity in Protein Families.Science.286(1999)295;
S.C.Trewick,T.F.Henshaw,R.P.Hansinger,T.Lindahl,B.Sedgwick.Oxidative demethylation by Escherichia coli AlkB directly reverts DNA base damage.Nature,419(2002)174;
Mallela M.G.Krishna,Yan Lin,Leland Mayne and S.Walter Englander.Intimate View of a Kinetic Protein Folding Intermediate:Residue-resolved Structure,Interactions,Stability,Folding and Unfolding Rates,Homogeneity.J.Mol.Biol.334(2003)501.
[171]M.B.Goodman,G.G.Ernstrom,D.S.Chelur,R.O.Hagan,C.A.Yao,M.Chalfie.MEC-2 regulates C.elegans DEG/ENaC channels needed for mechanosensation.Nature 415(2002)1039;
B.L.MacInnis,R.B.Campenot.Retrograde Support of Neuronal Survival Without Retrograde Transport of Nerve Growth Factor.Science 295(2002)1536.
[172]M.Abramowitz and I.Stegun.Handbook of mathematical functions.New York:Dover,1964;
D.F.Lawden.Elliptic functions and applications.New York:Spring verlag,1989.
[173]C.W.Cao,X.G.Geng,in:C.H.Gu,Y.S.Li,G.Z.Tu,(Eds.).Nonlinear Physics,Research Reports in Physics.Springer,Berlin,68(1990);
Y.Cheng,and Y.S.Li.The constraint of the KP equation and its special solutions.Phys.Lett.A,157(1991)22;
S.Y.Lou and L.L.Chen.Formally variable separation approach for nonintegrable models.J.Math.Phys.,40(1999)6491.
[174]C,Z.Qu,S.L.Zhang and Q.J.Zhang.Integrablity of Models arsing from motions of Plane curves.Z.Naturforsch,58a(2003)75;
S.L.Zhang,S.L.Lou and C.Z.Qu.Variable separation and exact solutions to generalized nonlinear diffusion equation.Chin.Phys.Lett.,19(2002)1741;
P.G.Est évez,C.Z.Qu and S.L.Zhang.Separation of variable a generalized porous medium equation with nonlinear source.J.Math.Anal.Appl.275(2002)44.
[175]S.L.Zhang and S.Y.Lou.Variable separation and derivative-dependent functional separable solutions to generalized KdV equation.Commum.Theor.Phys.,40(2003)401.
[176]A.D.D.Craik:Wave interactions and fluid flows.Cambridge university press,1985.
[177]A.Uthayakumar,K.Nakkeeran and K.Porsezian.Soliton solution of new(2+1) dimensional nonlinear partial differential equations.Chaos,Soliton and Fractals,10(1999)1513.
[178]M.Boiti,J.J.Pleon,M.Manna and F.Pempinelli.On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions.Inverse Problems,2(1986)271.
[179]C.L.Zheng,J.F.Zhang,Z.M.Sheng.Chaos and Fractals in a(2+1)-Dimensional Soliton System.Chin.Phys.Lett.,20(2003)331.
[180]S.Y.Lou and X.B.Hu.Infinitely many Lax pairs and symmetry constraints of the KP equation.J.Math.Phvs.,38(1997)6401.
[181]张金良,王跃明,王明亮,方宗德.变系数(2+1)维Broer-Kaup方程的精确解.原子与分子物理学报.20(2003)92.
[182]张解放,徐昌智,何宝钢.物理学报,变量分离法与变系数非线性薛定谔方程的求解探索.物理学报,53(2004)3652.
[183]K.W.Chow,A class of exact periodic solutions of nonlinear envelope equations.J.Math.Phys.,,36(1995)4125.
[184]K.W.Chow.periodic solutions for a system of four coupled nonlinear Schrodinger equations,Phys.Lett.A,285(2001)319;
K.W.Chow,Nakkeeran K,Malomed B A,Periodic waves in bimodal optical fibers,Optical Communications,219(2003)251;
K.W.Chow.Periodic waves for a system of coupled,higher order nonlinear Schrodinger equations with third order dispersion,Phys.Lett.A,308(2003)426.
[185]K.W.Chow.A class of douply periodic waves for nonlinear evolution equations,Wave Motion,35(2002)71;
K.W.Chow.Rational function representations of wave patterns in higherdemensional and discrete evolution equations,Phys.Lett.A,326(2004)401.
[186]A.Khare and U.Sukhatme.Cyclic identities involving Jacobi elliptic functions J.Math.Phys.43(2002)3798;
A.Khare and U.Sukhatme.Linear superposition in nonlinear equations.Phys.Rev.Lett.88(2002)244101.
A.Khare,et al.Cyclic identities for Jacobi elliptic and related functions.J.Math.Phys.44(2003)1822;
F.Cooper,et al.Periodic solutions of nonlinear equations obtained by linear superposition.J.Phys.A:Math.Gen.35(2002)10085.
[187]H.W.Sch ürmann,V.S.Serov,J.Nickel.Superposition in NonlinearWave and Evolution Equations.Int.J.Theor.Phys.,45(2006)1093.
[188]X.J.Lai and J.F.Zhang.A new class of periodic solutions to the(2+1)-dimensional breaking soliton equation.Chin.J.Phys.,42(2004)361;
J.F Zhang and X.J.Lai.An alternative construction of linear superposition periodic solutions to nonlinear equations.Phys.Lett.A,340(2005)188
[189]G.Georg and M.Boris.Stable two-dimensional parametric soitons in fluid system.Phys.Lett.A,248(1998)208.
[190]G.Georg and M.Boris.Parametric envelop soitons in coupled Korteweg-de Vries equations.Phys.Lett.A,227(1997)47.
[191]朱海平,郑春龙.(2+1)维广义Nizhnik-Novikov-Veselov系统的新严格解和复合波激发.物理学报.55(2006)4999.
马松华,方建平,朱海平.在Jacobi正弦周期波背景下的dromion孤立波及其演化.物理学报,56(2007)4319.
[192]A.Slunyaeva,C.Kharif,E.Pelinovsky,T.Talipova.Nonlinear wave focusing on water of finite depth.Physica D,173(2002)77;
C.M.Schober.Melnikov analysis and inverse spectral analysis of rogue waves in deep water.European Journal of Mechanics B,25(2006)602.
[193]C.Kharif,E.Pelinovsky.Physical mechanisms of the rogue wave phenomenon.European Journal of Mechanics B/Fluids,22(2003)603.
[194]A.Maccari.The Kadomtsev-Petviashvili equation as a source of integrable model equations.J.Math.Phys.,37(1996)6207.
[195]K.Porsezian.Painlevé analysis of new higher-dimensional soliton equation.J.Math.Phys.,38(1997)4675.