非线性Dirac方程族统一形式的Darboux变换和应用
|
摘要
本文主要研究Dirac方程族统一形式的Darboux变换及其应用,全文分四章:第一章,简要介绍孤立子理论以及与本文相关的Darboux变换的产生和发展过程.第二章,从Dirac谱问题出发,利用Lenard向量和零曲率方程推导出了Dirac方程族,其中典型方程为Dirac方程.第三章,基于Dirac方程族的Lax对,从理论上构造了整个方程族统一形式的Darboux变换.第四章,作为Darboux变换的一个应用,我们求出了Dirac方程的精确解.
In this paper,we focus on the Darboux transformation of the Dirac hierarchy and its application,in the first chapter,we simply introduce the soliton theory and the beginning and development process of the Darboux transformation related to this paper,in the second chapter,we start from the Dirac spectral problem,then derive a Dirac hierarchy by using Lenard operator pairs and the zero-curved equation. in the third chapter,an explicit and universal Darboux transformation for the whole hierarchy is constructed based on its Lax pair in theory.In the end,exact solutions of the Dirac hierarchy are constructed explicitly as an application of Darboux transformation.
In this paper,we focus on the Darboux transformation of the Dirac hierarchy and its application,in the first chapter,we simply introduce the soliton theory and the beginning and development process of the Darboux transformation related to this paper,in the second chapter,we start from the Dirac spectral problem,then derive a Dirac hierarchy by using Lenard operator pairs and the zero-curved equation. in the third chapter,an explicit and universal Darboux transformation for the whole hierarchy is constructed based on its Lax pair in theory.In the end,exact solutions of the Dirac hierarchy are constructed explicitly as an application of Darboux transformation.
引文
[1]Ablowitz M J,Carkson P A,Nonlinear Evolution and Inverse Scatter-ing.New York:Cambridge University Press,(1992),32-68
[2]Yoshimasa Matsuno.Bilinear Transformation Method[M],Academic Press,INC(1984)
[3]谷超豪等,孤立子理论与应用[M]浙江:浙江科技出版社,1990
[4]王明亮,非线性发展方程与孤立子[M],兰州:兰州大学出版社,1990
[5]李诩神,孤立子与可积系统『M],上海:上海科技教育出版社,1999
[6]谷超豪等.孤立子理论中的达布变换及其几何应用[M],上海:上海科学技术出版社.1999
[7]郭柏灵,苏凤秋.孤立子[M],辽宁:辽宁教育出版社,1998,172-294
[8]郭柏灵.庞小峰.孤立子[M],科学出版社.
[9]谷超豪.Darboux变换的可逆性、可换性和周期性[J],中国科大学报,1993,23(1):9-14
[10]Li Yishen,Ma Wenxiu,Zhang JinE.Darboux transformation of classical Boussinesq system and its solutions[J],Phy.Lett.A,2000,275:60-66
[11]Yi-shenli,JinE.Zhang.Darboux transformation of classical Boussinesq sys-tem and its multi-soliton solutions[J],Phys.Lett.A,2001,284:253-258
[12]LiYishen,Han Wen-ting.Deep reduetion of Darboux transformations to A 3x3 sPeetral Problem[J],Chin.Ann.ofMath.,2001,22B(2):171-176
[13]李栩神.一个特征值问题的达布变换与带外场的s-G方程的孤立子解[J],应用数学学报,1986,9(2):196-200
[14]Liyishett,Mawenxiu.Binary nonlinearization of AKNS sPectral Problem under higher-order symmetry constraints[J],chnao,solitons and Frae-tals,2000,11:697-710
[15]程艺,阿妹.积分型Daxboux变换[J],数学年刊,1999,20A(6):667-672
[16]Gu C H,et,al,soliton theory and Application,Shanghai:Shanghai Science and Technology Press.(1996)
[17]Gu C H,Hu H S and Zhou Z X,Darboux Transformation in soliton theory and its Geometric Applications,shanghai:shanghai science and technical Publishers,shanghai(1999)
[18]Gen X G,Tam Hon-Wah,Darboux Transformation and soliton solu-tions for the generalized nonlinear schrodinger Equation,J,Phys,Soc,Jpn mathbf(68)(1998)1508-1512
[19]Huxingbiao.A Powerful aPProach to generate new integrable sys-tems [J],J.Phys.A:mathgen.,1994,27:2497-2514
[20]陈登远,曾云波.Heisenbery型非线性发展方程[J],数学学报,1984,27(5):624-630
[21]陈登远,曾云波,李诩神.AKNs类发展方程的递推算子自身之间的转换算子[J],高校应用数学学报,1989,14(1):61-72
[22]Zhang Dajun,Chen dengyuan.The conservation laws of some diserete soliton systems[J],ehaos,solitons and Fraetals,2002,14:573-579
[23]Zhang dajun.The N-soliton solutions for the MKDV equation with self-consistent sources[J],J.phys.soeiety of Japan,2002,71(11):2649-2656
[24]Dajun Zhang and Dengyuan Chen.Harnilton structure of diserete soliton systems[J],Phys.A.math.,2002,35:7225-7241
[25]Wang mingliang,Zhou yubin.A nonlinear transformation of variant shal-low water wave equations and its applications[J1,Advances in mathe-maties,1999,28(1):71--75
[26]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法[J],物理学报,1998,47(3):353-363
[27]范恩贵.孤立子与可积系统[D],辽宁大连:大连理工大学,1998
[28]乔志军.一类具有C.Neumann约束的Hamilton系统及WKI孤子族[J],数学研究与评论,1993,13(3):377-352
[29]Ma Wenxiu.Symmetry constraint of MKDV equations by binary nonlin-earzation [J],Physiea.A,1995,219:1083-1091
[30]Ma wenxiu,Zhou ruguang.Binary nonlinearization of spectral Prob-lems of the Perturbation AKNS systems[JI,ehnao,solitons and Frae-tats,2002,13:1451-1463
[31]Ma wenxiu,Zhou ruguang.Adjoint symmetry constraints of multi-component AKNS equations[J],ehin.Ana.of math,2002,23B(3):373-384
[32]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[J],物理学报,2002,51(4):718-723
[33]刘式适,付遵涛,刘式达,赵强一类非线性方程的新周期解[J],物理学报,2002,51(1):10-15
[34]张卫国,马文秀.广义PC方程的显式精确孤子解[J],应用数学和力学,1999,20(6):625-633
[35]耿献国.变形Boussinesq方程的Lax对和Darboux变换解[J],应用数学学报,1988,113:324-328
[36]顾祝全一个KP型方程的Lax表示,Bachlund变换和无穷守恒律[J],科学通报,1989,2:86-89
[37]J.J.C.Nimmo.Darboux transformations for discrete sys-tems [J],ehnao,solitons and Fraetals,2000,11:115-120
[38]张桂戍.李志斌.段一士,非线性波方程的精确孤立子解[J],中国科学A 辑.2000.12:1103-1108
[39]Mirura M R,Backlund Transformastion.Berlin:Springer-Verlag,(1987)
[40]Wang M L,Zhou Y B,Li Z B,Application of a momogeneous balance method to exact solutions of nonlinear evolutions in mathematical physics,Phys,Lett.A216 1996 67
[41]Hereman W,Banerjee P P,Korpel A,Exact solitary wave solutions of non-linear evolution and wave equations using a direct algebraic method,J Phys.A:Math.Gen 19 1985 607
[42]Parkes E J,Duffy B R,An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations.Comp.Phys.Comm,98(1996) 288-300
[43]Li Z B,Shi H,Exact solutions for Belousov-Zhabotinskii reaction-diffusion system.Appl,Math.JCU 11B(1996)
[44]Zhou Z J,Li Z B,A Darboux transformation and new exact solutions for Broer-Kaup System,Acta Phys.sin53(2003) 262-265
[45]Fan E G Extended tanh-function method and its applications to nonlinear equations,textitPhys.Lett A.277(2000) 212-218
[46]Li Z B,Liu Y P,RATH:A maple package for finding traverlling solitary wave solutions to nonlinear equations,Comp,Phys.comm 148(2002) 256
[47]Liu Y p,Li Z B,An automated Jacobi elliptic function method for finding periodic wave solutions to nonlinear evolution equations Chin.phys.Lett.19(2002) 1228
[48]Yao R X,Li Z B New solitary wave solutions fpr nonlinear evolution equations.Chin,Physl1(2002) 864
[49]Yao R X,Li Z B,New exact solutions for three nonlinear Evolution Equations,Phys,Lett,A,297(2002) 196
[50]Li Z B,Exact solitary wave solutions and singuler solutions to the two-dimensional nonlinear dissipative-dispersive system,Acta math,Sci 22B (2002) 138
[51]Zhang S Q,Xu G Q,Li Z B,Gerneral Explicit solutions of a classical Boussi-nesq system,Chin,Phys,11(2002) 993
[52]Li Z B,Liu Y P,Wang M L,Exact solitary wave and soliton solutions of the fifth order model equation,Acta Math,Sci22B(2002) 138
[53]He Z M and Wang D L 2007 Chin.Phys.56 3088
[54]Frolve IS 1972 Sov.Math,Dokl,13 1468
[55]Grose H 1986 Phys,Rep,134 297
[56]Hinton D B jordan A K.Klaus M and Shaw J K 1991 J,Math.Phys.323015
[57]Ma W X 1997 Chin.Ann.Math.B 18 79
[2]Yoshimasa Matsuno.Bilinear Transformation Method[M],Academic Press,INC(1984)
[3]谷超豪等,孤立子理论与应用[M]浙江:浙江科技出版社,1990
[4]王明亮,非线性发展方程与孤立子[M],兰州:兰州大学出版社,1990
[5]李诩神,孤立子与可积系统『M],上海:上海科技教育出版社,1999
[6]谷超豪等.孤立子理论中的达布变换及其几何应用[M],上海:上海科学技术出版社.1999
[7]郭柏灵,苏凤秋.孤立子[M],辽宁:辽宁教育出版社,1998,172-294
[8]郭柏灵.庞小峰.孤立子[M],科学出版社.
[9]谷超豪.Darboux变换的可逆性、可换性和周期性[J],中国科大学报,1993,23(1):9-14
[10]Li Yishen,Ma Wenxiu,Zhang JinE.Darboux transformation of classical Boussinesq system and its solutions[J],Phy.Lett.A,2000,275:60-66
[11]Yi-shenli,JinE.Zhang.Darboux transformation of classical Boussinesq sys-tem and its multi-soliton solutions[J],Phys.Lett.A,2001,284:253-258
[12]LiYishen,Han Wen-ting.Deep reduetion of Darboux transformations to A 3x3 sPeetral Problem[J],Chin.Ann.ofMath.,2001,22B(2):171-176
[13]李栩神.一个特征值问题的达布变换与带外场的s-G方程的孤立子解[J],应用数学学报,1986,9(2):196-200
[14]Liyishett,Mawenxiu.Binary nonlinearization of AKNS sPectral Problem under higher-order symmetry constraints[J],chnao,solitons and Frae-tals,2000,11:697-710
[15]程艺,阿妹.积分型Daxboux变换[J],数学年刊,1999,20A(6):667-672
[16]Gu C H,et,al,soliton theory and Application,Shanghai:Shanghai Science and Technology Press.(1996)
[17]Gu C H,Hu H S and Zhou Z X,Darboux Transformation in soliton theory and its Geometric Applications,shanghai:shanghai science and technical Publishers,shanghai(1999)
[18]Gen X G,Tam Hon-Wah,Darboux Transformation and soliton solu-tions for the generalized nonlinear schrodinger Equation,J,Phys,Soc,Jpn mathbf(68)(1998)1508-1512
[19]Huxingbiao.A Powerful aPProach to generate new integrable sys-tems [J],J.Phys.A:mathgen.,1994,27:2497-2514
[20]陈登远,曾云波.Heisenbery型非线性发展方程[J],数学学报,1984,27(5):624-630
[21]陈登远,曾云波,李诩神.AKNs类发展方程的递推算子自身之间的转换算子[J],高校应用数学学报,1989,14(1):61-72
[22]Zhang Dajun,Chen dengyuan.The conservation laws of some diserete soliton systems[J],ehaos,solitons and Fraetals,2002,14:573-579
[23]Zhang dajun.The N-soliton solutions for the MKDV equation with self-consistent sources[J],J.phys.soeiety of Japan,2002,71(11):2649-2656
[24]Dajun Zhang and Dengyuan Chen.Harnilton structure of diserete soliton systems[J],Phys.A.math.,2002,35:7225-7241
[25]Wang mingliang,Zhou yubin.A nonlinear transformation of variant shal-low water wave equations and its applications[J1,Advances in mathe-maties,1999,28(1):71--75
[26]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法[J],物理学报,1998,47(3):353-363
[27]范恩贵.孤立子与可积系统[D],辽宁大连:大连理工大学,1998
[28]乔志军.一类具有C.Neumann约束的Hamilton系统及WKI孤子族[J],数学研究与评论,1993,13(3):377-352
[29]Ma Wenxiu.Symmetry constraint of MKDV equations by binary nonlin-earzation [J],Physiea.A,1995,219:1083-1091
[30]Ma wenxiu,Zhou ruguang.Binary nonlinearization of spectral Prob-lems of the Perturbation AKNS systems[JI,ehnao,solitons and Frae-tats,2002,13:1451-1463
[31]Ma wenxiu,Zhou ruguang.Adjoint symmetry constraints of multi-component AKNS equations[J],ehin.Ana.of math,2002,23B(3):373-384
[32]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[J],物理学报,2002,51(4):718-723
[33]刘式适,付遵涛,刘式达,赵强一类非线性方程的新周期解[J],物理学报,2002,51(1):10-15
[34]张卫国,马文秀.广义PC方程的显式精确孤子解[J],应用数学和力学,1999,20(6):625-633
[35]耿献国.变形Boussinesq方程的Lax对和Darboux变换解[J],应用数学学报,1988,113:324-328
[36]顾祝全一个KP型方程的Lax表示,Bachlund变换和无穷守恒律[J],科学通报,1989,2:86-89
[37]J.J.C.Nimmo.Darboux transformations for discrete sys-tems [J],ehnao,solitons and Fraetals,2000,11:115-120
[38]张桂戍.李志斌.段一士,非线性波方程的精确孤立子解[J],中国科学A 辑.2000.12:1103-1108
[39]Mirura M R,Backlund Transformastion.Berlin:Springer-Verlag,(1987)
[40]Wang M L,Zhou Y B,Li Z B,Application of a momogeneous balance method to exact solutions of nonlinear evolutions in mathematical physics,Phys,Lett.A216 1996 67
[41]Hereman W,Banerjee P P,Korpel A,Exact solitary wave solutions of non-linear evolution and wave equations using a direct algebraic method,J Phys.A:Math.Gen 19 1985 607
[42]Parkes E J,Duffy B R,An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations.Comp.Phys.Comm,98(1996) 288-300
[43]Li Z B,Shi H,Exact solutions for Belousov-Zhabotinskii reaction-diffusion system.Appl,Math.JCU 11B(1996)
[44]Zhou Z J,Li Z B,A Darboux transformation and new exact solutions for Broer-Kaup System,Acta Phys.sin53(2003) 262-265
[45]Fan E G Extended tanh-function method and its applications to nonlinear equations,textitPhys.Lett A.277(2000) 212-218
[46]Li Z B,Liu Y P,RATH:A maple package for finding traverlling solitary wave solutions to nonlinear equations,Comp,Phys.comm 148(2002) 256
[47]Liu Y p,Li Z B,An automated Jacobi elliptic function method for finding periodic wave solutions to nonlinear evolution equations Chin.phys.Lett.19(2002) 1228
[48]Yao R X,Li Z B New solitary wave solutions fpr nonlinear evolution equations.Chin,Physl1(2002) 864
[49]Yao R X,Li Z B,New exact solutions for three nonlinear Evolution Equations,Phys,Lett,A,297(2002) 196
[50]Li Z B,Exact solitary wave solutions and singuler solutions to the two-dimensional nonlinear dissipative-dispersive system,Acta math,Sci 22B (2002) 138
[51]Zhang S Q,Xu G Q,Li Z B,Gerneral Explicit solutions of a classical Boussi-nesq system,Chin,Phys,11(2002) 993
[52]Li Z B,Liu Y P,Wang M L,Exact solitary wave and soliton solutions of the fifth order model equation,Acta Math,Sci22B(2002) 138
[53]He Z M and Wang D L 2007 Chin.Phys.56 3088
[54]Frolve IS 1972 Sov.Math,Dokl,13 1468
[55]Grose H 1986 Phys,Rep,134 297
[56]Hinton D B jordan A K.Klaus M and Shaw J K 1991 J,Math.Phys.323015
[57]Ma W X 1997 Chin.Ann.Math.B 18 79