Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy
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摘要
In this paper, the authors define the noncommutative constrained KadomtsevPetviashvili(KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP(NcKP) hierarchy and multi-component noncommutative constrained KP(NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.
In this paper, the authors define the noncommutative constrained KadomtsevPetviashvili(KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP(NcKP) hierarchy and multi-component noncommutative constrained KP(NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.
In this paper, the authors define the noncommutative constrained KadomtsevPetviashvili(KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP(NcKP) hierarchy and multi-component noncommutative constrained KP(NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.
引文
[1]Fokas,A.S.and Santini,P.M.,The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schr¨odinger operator,Stud.Appl.Math.,75,1986,179-185.
[2]Fokas,A.S.and Santini,P.M.,Recursion operators and bi-Hamiltonian structures in multidimensions,II,Commun.Math.Phys.,116,1988,449-474.
[3]Matsukidaira,J.,Satsuma,J.and Strampp,W.,Conserved quantities and symmetries of KP hierarchy,J.Math.Phys.,31,1990,1426-1434.
[4]Fuchssteiner,B.and Fokas,A.S.,Symplectic structures,their B¨acklund transformation and hereditary symmetries,Phys.D.,4,1981,47-66.
[5]Olver,P.J.,Applications of Lie Groups to Differential Equations,Springer-Verlag,New York,1986.
[6]Santini,P.M.and Fokas,A.S.,Recursion operators and bi-Hamiltonian structures in multidimensions,I,Commun.Math.Phys.,115,1988,375-419.
[7]Strampp,W.and Oevel,W.,Recursion operators and Hamitonian structures in Sato theory,Lett.Math.Phys.,20,1990,195-210.
[8]Stephen,C.A.,Bi-Hamiltonian operators,integrable flows of curves using moving frames and geometric map equations,J.Phys.A:Math.Gen.,39,2006,2043-2072.
[9]Douglas,M.R.and Nekrasov,N.A.,Noncommutative field theory,Rev.Mod.Phys.,73,2002,p.977.
[10]Dickey,L.A.,Soliton Equations and Hamiltonian Systems,2nd ed.,World Scintific,Singapore,2003.
[11]Date,E.,Kashiwara,M.,Jimbo,M.,et al.,Transformation groups for soliton equations,Nonlinear Integrable Systems-Classical and Quantum Theory,Jimbo,M.and Miwa,T.(eds.),World Scientific,Singapore,1983,39-119.
[12]Cheng,Y.,Constraints of the Kadomtsev-Petviashvili hierarchy,J.Math.Phys.,33,1992,3774-3782.
[13]Gurses,M.,Karasu,A.and Sokolov,V.V.,On construction of recursion operators from Lax representation,J.Math.Phys.,40,1999,6473-6490.
[14]Loris,I.,Recursion operator for a constrained BKP system,Proceedings of the Workshop on Nonlinearity,Integrability and All That Twenty years after NEEDS’79,Boiti,M.,Martina,L.,Pempinelli,F.,et al.,(eds.),World Scientific,Singapore,1999,325-330.
[15]Li,C.Z.,Tian,K.L.,He,J.S.and Cheng,Y.,Recursion operator for a constrained CKP hierarchy,Acta Mathematica Scientia,Ser.B,31,2011,1295-1302.
[16]He,J.S.,Tu,J.Y.,Li,X.D.and Wang,L.H.,Explicit flow equations and recursion operator of the nc KPhierarchy,Nonlinearity,24,2011,2875-2890.
[17]Hamanaka,M.,Noncommutative integrable systems and quasidetermiants,Nonlinear and Modern Mathematical Physics,AIP Conf.Proc.,1212,Amer.Inst.Phys.,Melville,NY,2010,122-135.
[18]Sooman,G.M.,Soliton Solutions of Noncommutative Integrable Systems,University of Glasgow,2009.
[2]Fokas,A.S.and Santini,P.M.,Recursion operators and bi-Hamiltonian structures in multidimensions,II,Commun.Math.Phys.,116,1988,449-474.
[3]Matsukidaira,J.,Satsuma,J.and Strampp,W.,Conserved quantities and symmetries of KP hierarchy,J.Math.Phys.,31,1990,1426-1434.
[4]Fuchssteiner,B.and Fokas,A.S.,Symplectic structures,their B¨acklund transformation and hereditary symmetries,Phys.D.,4,1981,47-66.
[5]Olver,P.J.,Applications of Lie Groups to Differential Equations,Springer-Verlag,New York,1986.
[6]Santini,P.M.and Fokas,A.S.,Recursion operators and bi-Hamiltonian structures in multidimensions,I,Commun.Math.Phys.,115,1988,375-419.
[7]Strampp,W.and Oevel,W.,Recursion operators and Hamitonian structures in Sato theory,Lett.Math.Phys.,20,1990,195-210.
[8]Stephen,C.A.,Bi-Hamiltonian operators,integrable flows of curves using moving frames and geometric map equations,J.Phys.A:Math.Gen.,39,2006,2043-2072.
[9]Douglas,M.R.and Nekrasov,N.A.,Noncommutative field theory,Rev.Mod.Phys.,73,2002,p.977.
[10]Dickey,L.A.,Soliton Equations and Hamiltonian Systems,2nd ed.,World Scintific,Singapore,2003.
[11]Date,E.,Kashiwara,M.,Jimbo,M.,et al.,Transformation groups for soliton equations,Nonlinear Integrable Systems-Classical and Quantum Theory,Jimbo,M.and Miwa,T.(eds.),World Scientific,Singapore,1983,39-119.
[12]Cheng,Y.,Constraints of the Kadomtsev-Petviashvili hierarchy,J.Math.Phys.,33,1992,3774-3782.
[13]Gurses,M.,Karasu,A.and Sokolov,V.V.,On construction of recursion operators from Lax representation,J.Math.Phys.,40,1999,6473-6490.
[14]Loris,I.,Recursion operator for a constrained BKP system,Proceedings of the Workshop on Nonlinearity,Integrability and All That Twenty years after NEEDS’79,Boiti,M.,Martina,L.,Pempinelli,F.,et al.,(eds.),World Scientific,Singapore,1999,325-330.
[15]Li,C.Z.,Tian,K.L.,He,J.S.and Cheng,Y.,Recursion operator for a constrained CKP hierarchy,Acta Mathematica Scientia,Ser.B,31,2011,1295-1302.
[16]He,J.S.,Tu,J.Y.,Li,X.D.and Wang,L.H.,Explicit flow equations and recursion operator of the nc KPhierarchy,Nonlinearity,24,2011,2875-2890.
[17]Hamanaka,M.,Noncommutative integrable systems and quasidetermiants,Nonlinear and Modern Mathematical Physics,AIP Conf.Proc.,1212,Amer.Inst.Phys.,Melville,NY,2010,122-135.
[18]Sooman,G.M.,Soliton Solutions of Noncommutative Integrable Systems,University of Glasgow,2009.