Kaup-Newell族的非线性双可积耦合及其自相容源
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摘要
本文基于新的非半单矩阵Lie代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出Kaup-Newell族的非线性双可积耦合及其Hamilton结构.最后利用源生成理论建立新的公式,并导出带自相容源Kaup-Newell族的非线性双可积耦合方程.
Based on new non-semisimple matrix Lie algebras, we introduce the general method of constructing the nonlinear bi-integrable couplings of soliton hierarchy. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. As an application, we give the nonlinear bi-integrable couplings of Kaup-Newell hierarchy and its Hamiltonian structures. Finally,we set up a set of new formulas using the theory of source, and the nonlinear bi-integrable couplings of Kaup-Newell hierarchy with self-consistent sources is derived based on the new formulas.
Based on new non-semisimple matrix Lie algebras, we introduce the general method of constructing the nonlinear bi-integrable couplings of soliton hierarchy. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. As an application, we give the nonlinear bi-integrable couplings of Kaup-Newell hierarchy and its Hamiltonian structures. Finally,we set up a set of new formulas using the theory of source, and the nonlinear bi-integrable couplings of Kaup-Newell hierarchy with self-consistent sources is derived based on the new formulas.
引文
[1]MA Wenxiu.Multi-component bi-Hamiltonian Dirac integrable equations[J].Chaos Solitons Fract.,2009,39(1):282-287.
[2]ZHANG Yufeng.A generalized multi-component Glachette-Johnson(GJ)hierarchy and its integrable coupling system[J].Chaos Solitons Fract.,2004,21(2):305-310.
[3]ZHANG Yufeng,YAN Qingyou.Integrable couplings of the hierarchies of evolution equations[J].Chaos Solitons Fract.,2003,16(2):263-269.
[4]YU Fajun,LI Li.An integrable couplings of Dirac soliton hierarchy with self-consistent sources[J].Appl.Math.Comput.,2009,207(1):171-177.
[5]XIA Tiecheng,FAN Engui.The multicomponent generalized Kaup-Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions[J].J.Math.Phys.,2005,46(4):043510.
[6]XIA Tiecheng,ZHANG Gailian,FAN Engui.New integrable couplings of generalized Kaup-Newell hierarchy and its Hamiltonian structures[J].Commun.Theor.Phys.,2011,56(1):1-4.
[7]MA Wenxiu,FUSHSSTEINER B.The bi-Hamiltonian structure of the perturbation equations of the Kd V hierarchy[J].Phys.Lett.A,1996,213(1):49-55.
[8]MA Wenxiu.Integrable theory of the perturbation equations[J].Chaos Solitons Fract.,1996,7(8):1227-1250.
[9]MA Wenxiu.Nonlinear continuous integrable Hamiltonian couplings[J].Appl.Math.Comput.,2011,217(17):7238-7244.
[10]ZHANG Yufeng,MEI Jianqin.Lie algebras and integrable systems[J].Commun.Theor.Phys.,2012,57(6):1012-1022.
[11]ZHANG Yufeng.Lie algebras for constructing nonlinear integrable couplings[J].Commun.Theor.Phys.,2011,56(5):805-812.
[12]YU Fajun,LI Li.A nonlinear integrable couplings of C-Kd V soliton hierarchy and its infinite conservation laws[J].Appl.Math.Comput.,2014,233(1):413-420.
[13]WEI HAnyu,XIA Tiecheng.The generalized fractional trace variational identity and fractional integrable couplings of Kaup-Newell hierarchy[J].J.Math.Phys.,2014,55(8):083501.
[14]MA Wenxiu.Soliton,positon and negaton solutions to a Schr¨odinger self-consistent source equation[J].J.Phys.Soc.Jpn.,2003,72(12):3017-3019.
[15]YU Fajun.A non-isospectral integrable couplings of Volterra lattice hierarchy with self-consistent sources[J].Appl.Math.Comput.,2009,215(3):1217-1223.
[16]XIA Tiecheng.Two new integrable couplings of the soliton hierarchies with self-consistent sources[J].Chin.Phys.B,2010,19(10):100303.
[17]MEL’NIKOV V K.Integration of the nonlinear Schr¨odinger equation with a source[J].Inverse Probl.,1992,8(1):133-147.
[18]DOKTROV E V,VLASOV R A.Optical solitons in media with resonant and non-resonant selffocusing nonlinearities[J].Opt.Acta.,1983,30(2):223-232.
[19]MA Wenxiu,CHEN Min.Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras[J].J.Phys.A:Math.Gen.,2006,39(34):10787-10801.
[20]TAO Sixing,XIA TIecheng,SHI Hui.Super-KN hierarchy and its super-Hamiltonian structure[J].Commun.Theor.Phys.,2011,55(3):391-395.
[21]MA Wenxiu,MENG Jinghan,ZHANG Mengshu.Nonlinear bi-integrablecouplings with Hamiltonian structures[J].Math.Comput.Simulat.,2016,127(2):166-177.
[22]TU Guizhang.An extension of a theoremon gradients of conserved densities of integrable systems[J].Northeastern Math.J.,1990,6(1):26-32.
[2]ZHANG Yufeng.A generalized multi-component Glachette-Johnson(GJ)hierarchy and its integrable coupling system[J].Chaos Solitons Fract.,2004,21(2):305-310.
[3]ZHANG Yufeng,YAN Qingyou.Integrable couplings of the hierarchies of evolution equations[J].Chaos Solitons Fract.,2003,16(2):263-269.
[4]YU Fajun,LI Li.An integrable couplings of Dirac soliton hierarchy with self-consistent sources[J].Appl.Math.Comput.,2009,207(1):171-177.
[5]XIA Tiecheng,FAN Engui.The multicomponent generalized Kaup-Newell hierarchy and its multicomponent integrable couplings system with two arbitrary functions[J].J.Math.Phys.,2005,46(4):043510.
[6]XIA Tiecheng,ZHANG Gailian,FAN Engui.New integrable couplings of generalized Kaup-Newell hierarchy and its Hamiltonian structures[J].Commun.Theor.Phys.,2011,56(1):1-4.
[7]MA Wenxiu,FUSHSSTEINER B.The bi-Hamiltonian structure of the perturbation equations of the Kd V hierarchy[J].Phys.Lett.A,1996,213(1):49-55.
[8]MA Wenxiu.Integrable theory of the perturbation equations[J].Chaos Solitons Fract.,1996,7(8):1227-1250.
[9]MA Wenxiu.Nonlinear continuous integrable Hamiltonian couplings[J].Appl.Math.Comput.,2011,217(17):7238-7244.
[10]ZHANG Yufeng,MEI Jianqin.Lie algebras and integrable systems[J].Commun.Theor.Phys.,2012,57(6):1012-1022.
[11]ZHANG Yufeng.Lie algebras for constructing nonlinear integrable couplings[J].Commun.Theor.Phys.,2011,56(5):805-812.
[12]YU Fajun,LI Li.A nonlinear integrable couplings of C-Kd V soliton hierarchy and its infinite conservation laws[J].Appl.Math.Comput.,2014,233(1):413-420.
[13]WEI HAnyu,XIA Tiecheng.The generalized fractional trace variational identity and fractional integrable couplings of Kaup-Newell hierarchy[J].J.Math.Phys.,2014,55(8):083501.
[14]MA Wenxiu.Soliton,positon and negaton solutions to a Schr¨odinger self-consistent source equation[J].J.Phys.Soc.Jpn.,2003,72(12):3017-3019.
[15]YU Fajun.A non-isospectral integrable couplings of Volterra lattice hierarchy with self-consistent sources[J].Appl.Math.Comput.,2009,215(3):1217-1223.
[16]XIA Tiecheng.Two new integrable couplings of the soliton hierarchies with self-consistent sources[J].Chin.Phys.B,2010,19(10):100303.
[17]MEL’NIKOV V K.Integration of the nonlinear Schr¨odinger equation with a source[J].Inverse Probl.,1992,8(1):133-147.
[18]DOKTROV E V,VLASOV R A.Optical solitons in media with resonant and non-resonant selffocusing nonlinearities[J].Opt.Acta.,1983,30(2):223-232.
[19]MA Wenxiu,CHEN Min.Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras[J].J.Phys.A:Math.Gen.,2006,39(34):10787-10801.
[20]TAO Sixing,XIA TIecheng,SHI Hui.Super-KN hierarchy and its super-Hamiltonian structure[J].Commun.Theor.Phys.,2011,55(3):391-395.
[21]MA Wenxiu,MENG Jinghan,ZHANG Mengshu.Nonlinear bi-integrablecouplings with Hamiltonian structures[J].Math.Comput.Simulat.,2016,127(2):166-177.
[22]TU Guizhang.An extension of a theoremon gradients of conserved densities of integrable systems[J].Northeastern Math.J.,1990,6(1):26-32.