约束离散KP系列的对称代数及行列式解
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摘要
本文从对称性与解析性两方面来研究约束离散KP(cdKP)系列.首先,本文定义了离散KP(dKP)系列的ghost对称,并得至(?)ghost流对波函数和(共轭)波函数的作用与谱表示；构造了约束离散KP(cdKP)系列的附加对称流,并证明了其形成Virasoro代数.并得到此附加对称流对cdKP系列的τ函数的作用.其次,本文构造了cdKP系列的规范变换,并由此得至(?)JcdKP系列的Wronskian解.
In this paper, symmetry algebra and the analyticity of the discrete KP(dKP) hierarchy are discussed. Firstly, ghost flow of the dKP hierar-chy is defined, then the ghost flow on eigenfunction(adjoint eigenfunction) and spectral representation of Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. The ghost flow on tau function and another kind of proof of ASvM formula of the discrete KP hierarchy are given. We constructed the additional symmetries of one-component constrained discrete KP (cdKP) hierarchy, and then proved that the symmetry flows formed the Virasoro algebra. The action of the Virasoro symmetry on the tau function of the constrained discrete KP (cdKP) hierarchy is derived. The gauge transfor-mation of the constrained discrete KP hierarchy is constructed explicitly by the suitable choice of the generating functions. Under the m-step successive gauge transformation Tm, we give the transformed (adjoint) eigenfunctions and the τ-function of the transformed Lax operator of the cdKP hierarchy. We also analyzed the Wronskian solution of the eigenfunction of the cdKP hierarchy.
In this paper, symmetry algebra and the analyticity of the discrete KP(dKP) hierarchy are discussed. Firstly, ghost flow of the dKP hierar-chy is defined, then the ghost flow on eigenfunction(adjoint eigenfunction) and spectral representation of Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. The ghost flow on tau function and another kind of proof of ASvM formula of the discrete KP hierarchy are given. We constructed the additional symmetries of one-component constrained discrete KP (cdKP) hierarchy, and then proved that the symmetry flows formed the Virasoro algebra. The action of the Virasoro symmetry on the tau function of the constrained discrete KP (cdKP) hierarchy is derived. The gauge transfor-mation of the constrained discrete KP hierarchy is constructed explicitly by the suitable choice of the generating functions. Under the m-step successive gauge transformation Tm, we give the transformed (adjoint) eigenfunctions and the τ-function of the transformed Lax operator of the cdKP hierarchy. We also analyzed the Wronskian solution of the eigenfunction of the cdKP hierarchy.
引文
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[12]M. H. Tu, On the BKP hierarchy:additional symmetries, Fay identity and Adler-Shiota-van Moerbeke formula, Lett. Math. Phys.81,91(2007),93-105.
[13]J. S. He, K. L. Tian, A. Foerster and W. X. Ma, Additional Symmetries and String Equation of the CKP Hierarchy, Lett. Math. Phys.81(2007),119-134.
[14]J. P. Cheng, J. S. He and S. Hu, The "Ghost" Symmetry of the BKP hier-archy, J. Math. Phys 51(2010),053514.
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[22]Y. Cheng and Y. S. Li, The constraint of the Kadomtsev-Petviashvili equa-tion and its special solutions, Phys. Lett. A157 (1991),22-26.
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[26]L. A. Dickey, On additional symmetries of the KP hierarchy and Sato's Backlund transformation, Commun. Math. Phys.,167(1995),227-233.
[27]H. Aratyn, E. Nissimov and S. Pacheva, Virasoro symmetry of constrained KP Hierarchies, Phys. Lett. A 228(1997),164-175.
[28]K. L. Tian, J. S. He, J. P. Cheng and Y. Cheng, Additional symmetries of constrained CKP and BKP hierarchies, Sci. China Math.54(2011),257-268.
[29]H. F. Shen and M. H. Tu, On the constrained B-type Kadomtsev-Petviashvili hierarchy:Hirota bilinear equations and Virasoro symmetry, J. Math. Phys. 52(2011),032704.
[30]K. Ueno and K. Takasaki, Toda Lattice Hierarchy, Adv. Stud. Pure Math. 4(1984),1-95.
[31]K. Takasaki and T. Takebe, SDIFF(2) KP Hierarchy, Int. J. Mod. Phys. A 7(Suppl.1B)(1992),889-922.
[32]K. Takasaki and T. Takebe, Integrable hierarchies and dispersionless limit, Rev. Math. Phys.7(1995),743-808.
[33]B Dubrovin and Y. Zhang, Virasoro symmetries of the extended Toda hier-archy, Communications in mathematical physics,250(2004),161-193.
[34]J. P. Cheng, K. L. Tian and J. S. He, The additional symmetries for the BTL and CTL hierarchies, J. Math. Phys.52(2011),053515.
[35]F. Nijhoff and H. Capel, Acta Appl. Math.39(1995),133-158.
[36]B. A. Kupershimidt, Discrete Lax equations and differential-difference cal-culus, Asterisque,123(1985),1-212.
[37]R. Hirota, Discrete analogue of a generalized Toda equation, J. Phys. Soc. Jpn.50(1981) 3785-3791.
[38]L. Haine and P. Iliev, Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Notices 2000,281-323(2000).
[39]C. Z. Li, J. P. Cheng, K. L. Tian, M. H. Li and J. S. He, Ghost symmetry of the discrete KP hierarchy, arXiv.1201.4419.
[40]S. W. Liu and Y. Cheng, Sato Backlund transformation, additional symmtries and ASvM formular for the discrete KP hierarchy, J. Phys. A: Math. Theor.,43(2010),135202.
[41]X. L. Sun, D. J. Zhang, X. Y. Zhu and D. Y. Chen, Symmetries and Lie algebra of the differential-difference Kadomstev-Petviashvili hierarchy, Mod. Phys. Lett. B,24(2010),1033-1042, also see arXiv:0908.0382.
[42]W. Pu, L. Huang, K. M. Tamizhmani and D. J. Zhang, Integrable properties of the differential-difference Kadomtsev-Petviashvili hierarchy and continu-um limits, arXiv:1211.3585.
[43]S. Kanaga Vel and K. M. Tamizhmani, Lax Pairs, Symmetries and Conser-vation Laws of a Differential-Difference Equation-Sato's Approach, Chaos, Solitons and Fractals,8(1997) 917-931.
[44]S. W. Liu, Y. Cheng and J. S. He, The determinant representation of the gauge transformation for the discrete KP hierarchy, Sci. China Math., 53(2010),1195-1206.
[45]R. Willox, T. Tokihiro, I. Loris and J. Satsuma, The fermionic approach to Darboux transformations, Inverse Problems,14(1998),745-762.
[46]R. Willox, T. Tokihiro, J. Satsuma, Darboux and binary Darboux trans-formations for the nonautonomous discrete KP equation, J. Math. Phys. 38(1997),6455-6469.
[47]Y. Q. Yao, X. J. Liu and Y. B. Zeng, A new extended discrete KP hierar-chy and generalized dressing method, J. Phys. A:Math. Theor.,42(2009), 454026.
[48]M. Adler and P. van Moerbeke, Vertex operator solutions to the discrete KP-hierarchy, Commun. Math. Phys.,203(1999),185-210.
[49]L. A. Dickey. Modified KP and discrete KP, Lett. Math. Phys.48(1999), 277-89.
[50]W. Oevel, Darboux theorems and Wronskian formulas for integrable systems I:constrained KP flows, Physica A,195(1993),533-576.
[51]H. Aratyn, E. Nissimov, S. Pacheva, A new "dual" symmetry structure of the KP hierarchy, Phys. Lett. A,244(1998),245-255.
[52]H. Aratyn, E. Nissimov, S. Pacheva, Method of squared eigenfunction poten-tials in integrable hierarchies of KP type, Comm. Math. Phys.,193(1998), 493-525.
[53]W. Oevel and W. Schief, Squared eigenfunctions of the (modified) KP hier-archy and scattering Problems of Loewner type, Rev. Math. Phys.,6(1994), 1301-1338.
[54]W. Oevel. Darboux transformations for integrable lattice systems. Nonlinear Physics:theory and experiment. E. Alfinito, L. Martina and F. Pempinelli (eds), World Scientific, Singapore,1996,233-240.
[55]W. Oevel and S. Carillo, Squared eigenfunction symmetries for soliton equa-tions:Part I, J. Math. Anal. Appl.,217(1998),161-178.
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