具有广义发展项的一类K（m，n）方程的行波解分支及动力学研究
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摘要
本文利用下面动力系统分支理论和方法研究了具有广义发展项的k(m，n)方程(?)的特殊情况(M=3，n=2，，L≥2)的相图分支和全局动力学行为，并给出部分情况下的精确行波解的完全显式参数表达文中利用时间尺度变换，把具有广义发展项的一类k(m，n)方程的奇异行波系统转化为一个正则系统，再运用动力系统分支理论给出正则系统的相图分支和动力学性质，通过对正则系统的研究揭示奇异行波系统的行波解的存在性及其动力学行为，并求出部分情况下的精确行波解，进一步说明周期解的存在性
In this paper, by using bifurcation theory and methods of plane dynamic system, we investigate thebifurcation and dynamical behavior of some special cases of a class of K(m, n) equations, i.e. (ul)t +aumux +(?), n = 2, l≥2. We obtain some exact explicit parametric representations oftraveling wave solutions by using time scaling transformation. We convert K(m, n) equation into a regularsystem, then the bifurcation and dynamical behaviors of regular system is discussed using bifurcation theoryof dynamic system. By discussing regular system, we find the existence of traveling wave solution anddynamical behavior of the singular traveling wave system. Meanwhile we get some exact traveling waveand the existence of periodic solution.
In this paper, by using bifurcation theory and methods of plane dynamic system, we investigate thebifurcation and dynamical behavior of some special cases of a class of K(m, n) equations, i.e. (ul)t +aumux +(?), n = 2, l≥2. We obtain some exact explicit parametric representations oftraveling wave solutions by using time scaling transformation. We convert K(m, n) equation into a regularsystem, then the bifurcation and dynamical behaviors of regular system is discussed using bifurcation theoryof dynamic system. By discussing regular system, we find the existence of traveling wave solution anddynamical behavior of the singular traveling wave system. Meanwhile we get some exact traveling waveand the existence of periodic solution.
引文
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[3]冯大河，非线性波方程的精确解与分支问题研究[D]：[博士学位论文]昆明：昆明理工大学，2007
[4]He T L，Bifurcation of traveling wave solutions of(2+1)dimensional Konopelchenko—Dubrovsky equations，Appl Math Comput,2008，204(2)：773—783
[5]Li J B，Dynamical understanding of loop soliton solutions for several nonlinear wave equations[J]Sci China Ser A，2007，50(6)：773—785
[6]Li J B，Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equation[J]J DynDiffer equ，2008，20(4)：909—922
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[8]Li JB，LiuZR Travelingwave solutionsfor a elass ofnonlineardisPersive equations[J]ChineseAnnMathB，2002，23(3)：397—41 8
[9]Li JB，Shen J W Travellingwave solutionsin amodel ofthehelixPolyPePtide chains[J]Chaos SolitonFract，2004，20(4)：827—841
[10]Li J B，Zhang J X，Bifurcations of travelling wave solutions for the generalization form of the modified KdV equation[J]Chaos SolitonFracL 2004，21(4)：899—913
[11]Li JB，ZhangYExactM／Vv2shape solitarywave solutiondeterminedby a singulartravelingwave equation[J]NonlinearAnal—Real，2009，10(3)：1797—1 802
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[13]中建伟，非线性波方程行波解分岔及其动力学行为研究[D]：[博士学位论文]西安：西北工业大学，2006
[14]TrikiH，WhzwazAM，Soliton solutionsfor(2+1)一dimensionaland(3+1)一dimensionalK(m，n)equation[J]ApplMath Comput，2009，217(4)：1733—1740
[15]Ko~eg D J，de Vries G，On the change of form of long waves advancing in a rectangular canal，and on a new type of long stationay waves[J]Philos Mag，1 895，39(5)：422—443
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[17]Chen A Y Li J B，Single peak solitary wave solutions for the osmosis K(2，2)equation under inhomogeneous boundary condition J Math Anal Appl，2010，369(2)：758—766
[18]Tang SQ，LiMBifurcations oftravellingwave solutionsin a class ofgeneralizedKdV equation[J]ApplMath Comput，2006，177(2)：589—596
[19]XuCH，TianLX，The bifurcation andpeakonforK(2，2)equationwith osmosis dispersion[J]Chaos SolitonFract，2009，40(2)：893—901
[20]Yan Z Y Modified nonlinearly dispersive mK(m，n，k)equmions：II Jacobi elliptic function solutions[J]2003，153(1)：1-16
[21]Zhu Y G，Tong，Temuer Chaolu，New exact solitary—wave solutions for the K(2，2，1)and K(3，3，1)equa—tions[J]Chaos SolitonFract，2007，33(4)：1411-1416
[22]Biswas A，1-soliton solution ofthe K(m，n)equation with generalized evolution[J]Phys Lett A，2008，372(25)：4601—4602
[23]Li J B，Dai H H，On the study of singular nonlinear traveling wave equmions：dynamical system approach[M]Beijing：Science Press，2007 19—156
[24]ByrdPEFridmanMD，HandbookofEllipticIntegrslsForEngineers andSciensists[M]Berlin：Springer，1971
[25]刘式达，刘式适，叶其孝，非线性演化方程的显式行波解[J]数学的实践与认识，1998，28(4)：289—301
[26]刘式达，刘式适，孤立波与同宿轨道[J]力学与实践，1991，13(4)：9-15
[27]李继彬，两非线性波方程真圈解的存在性和破缺性质[J]应用数学和力学，2009，30(5)：505—514
[2]王振东，孤立子与孤立波[J]力学与实践，2005，27(5)：86—88
[3]冯大河，非线性波方程的精确解与分支问题研究[D]：[博士学位论文]昆明：昆明理工大学，2007
[4]He T L，Bifurcation of traveling wave solutions of(2+1)dimensional Konopelchenko—Dubrovsky equations，Appl Math Comput,2008，204(2)：773—783
[5]Li J B，Dynamical understanding of loop soliton solutions for several nonlinear wave equations[J]Sci China Ser A，2007，50(6)：773—785
[6]Li J B，Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equation[J]J DynDiffer equ，2008，20(4)：909—922
[7]Li J B，Liu Z R，Smooth and non—smooth traveling waves in an onlinearly disPersive equation[J]Appl MathModel，2000，25(1)：41—56
[8]Li JB，LiuZR Travelingwave solutionsfor a elass ofnonlineardisPersive equations[J]ChineseAnnMathB，2002，23(3)：397—41 8
[9]Li JB，Shen J W Travellingwave solutionsin amodel ofthehelixPolyPePtide chains[J]Chaos SolitonFract，2004，20(4)：827—841
[10]Li J B，Zhang J X，Bifurcations of travelling wave solutions for the generalization form of the modified KdV equation[J]Chaos SolitonFracL 2004，21(4)：899—913
[11]Li JB，ZhangYExactM／Vv2shape solitarywave solutiondeterminedby a singulartravelingwave equation[J]NonlinearAnal—Real，2009，10(3)：1797—1 802
[12]仇钊成，K(n，--n,2n)方程的行波解及其动力学性质的分析[D]：[硕士学位论文]上海：华东师范大学，2006
[13]中建伟，非线性波方程行波解分岔及其动力学行为研究[D]：[博士学位论文]西安：西北工业大学，2006
[14]TrikiH，WhzwazAM，Soliton solutionsfor(2+1)一dimensionaland(3+1)一dimensionalK(m，n)equation[J]ApplMath Comput，2009，217(4)：1733—1740
[15]Ko~eg D J，de Vries G，On the change of form of long waves advancing in a rectangular canal，and on a new type of long stationay waves[J]Philos Mag，1 895，39(5)：422—443
[16]Rosenau F，Hyman JMCompacton：solitonswithfinitewavelength[J]PhysRevLet*，1993，70(5)：564—567
[17]Chen A Y Li J B，Single peak solitary wave solutions for the osmosis K(2，2)equation under inhomogeneous boundary condition J Math Anal Appl，2010，369(2)：758—766
[18]Tang SQ，LiMBifurcations oftravellingwave solutionsin a class ofgeneralizedKdV equation[J]ApplMath Comput，2006，177(2)：589—596
[19]XuCH，TianLX，The bifurcation andpeakonforK(2，2)equationwith osmosis dispersion[J]Chaos SolitonFract，2009，40(2)：893—901
[20]Yan Z Y Modified nonlinearly dispersive mK(m，n，k)equmions：II Jacobi elliptic function solutions[J]2003，153(1)：1-16
[21]Zhu Y G，Tong，Temuer Chaolu，New exact solitary—wave solutions for the K(2，2，1)and K(3，3，1)equa—tions[J]Chaos SolitonFract，2007，33(4)：1411-1416
[22]Biswas A，1-soliton solution ofthe K(m，n)equation with generalized evolution[J]Phys Lett A，2008，372(25)：4601—4602
[23]Li J B，Dai H H，On the study of singular nonlinear traveling wave equmions：dynamical system approach[M]Beijing：Science Press，2007 19—156
[24]ByrdPEFridmanMD，HandbookofEllipticIntegrslsForEngineers andSciensists[M]Berlin：Springer，1971
[25]刘式达，刘式适，叶其孝，非线性演化方程的显式行波解[J]数学的实践与认识，1998，28(4)：289—301
[26]刘式达，刘式适，孤立波与同宿轨道[J]力学与实践，1991，13(4)：9-15
[27]李继彬，两非线性波方程真圈解的存在性和破缺性质[J]应用数学和力学，2009，30(5)：505—514