Abstract
In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction (adjoint eigenfunction) and the spectral representation of its Baker–Akhiezer function and adjoint Baker–Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.
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References
Kupershimidt, B.A.: Discrete Lax equations and differential-difference calculus. Asterisque 123, 1–212 (1985)
KanagaVel, S., Tamizhmani, K.M.: Lax Pairs, Symmetries and Conservation Laws of a Differential-Difference Equation-Sato’s Approach. Chaos, Solitons and Fractals 8, 917–931 (1997)
Haine, L., Iliev, P.: Commutative rings of difference operators and an adelic flag manifold. Int. Math. Res. Not. 6, 281–323 (2000)
Date, E., Kashiwara, M., Jimbo M., Miwa, T.: Transformation groups for soliton equations nonlinear integrable systems-classical and quantum theory (Kyoto 1981), In: Jimbo M , Miwa T (eds) Singapore: World Scientific, pp. 39–119
Dickey, L.A.: Soliton equations and hamiltonian systems, 2nd edn. River Edge World Scientific, Singapore, NJ (2003)
Liu, S.W., Cheng, Y., He, J.S.: The determinant representation of the gauge transformation for the discrete KP hierarchy. Sci China Math 53, 1195–1206 (2010)
Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grasmann manifolds. In: Nonlinear PDE in Applied Science, vol. 5, pp. 259–271. U.S.-Japan Seminar, Tokyo, Lecture Notes Num. Appl. Anal. (1982)
Liu, S.W., Cheng, Y.: Sato backlund transformation, additional symmtries and ASvM formular for the discrete KP hierarchy. J. Phys. A: Math. Theor., 43, 135202 (2010)
Willox, R., Tokihiro, T., Loris, I., Satsuma, J.: The fermionic approach to darboux transformations. Inverse Probl 14, 745–762 (1998)
Willox, R., Tokihiro, T., Satsuma, J.: Darboux and binary Darboux transformations for the nonautonomous discrete KP equation. J. Math. Phys. 38, 6455–6469 (1997)
Yao, Y.Q., Liu, X.J., Zeng, Y.B.: A new extended discrete KP hierarchy and generalized dressing method. J. Phys. A: Math. Theor. 42, 454026 (2009)
Sun, X.L., Zhang, D.J., Zhu, X.Y., Chen, D.Y.: Symmetries and Lie algebra of the differential-difference Kadomstev-Petviashvili hierarchy. Mod. Phys. Lett. B 24, 1033–1042 (2010). arXiv:0908.0382
Adler, M., van Moerbeke, P.: Vertex operator solutions to the discrete KP-hierarchy. Commun. Math. Phys. 203, 185–210 (1999)
Dickey, L.A.: Modified KP and discrete KP. Lett. Math. Phys. 48, 277–89 (1999)
Oevel, W.: Darboux theorems and Wronskian formulas for integrable systems I: constrained KP flows. Phy A 195, 533–576 (1993)
Oevel, W., Schief, W.: Squared eigenfunctions of the (modified) KP hierarchy and scattering problems of loewner type. Rev. Math. Phys. 6, 1301–1338 (1994)
Oevel, W.: Darboux transformations for integrable lattice systems. In: Alfinito, E., Martina L., Pempinelli F. (eds) Nonlinear Physics: Theory and Experiment, pp. 233–240. World Scientific, Singapore (1996)
Aratyn, H., Nissimov, E., Pacheva, S.: Virasoro symmetry of constrained KP hierarchies. Phys. Lett. A 228, 164–175 (1997)
Aratyn, H., Nissimov, E., Pacheva, S.: A new “dual” symmetry structure of the KP hierarchy. Phys. Lett. A 244, 245–255 (1998)
Aratyn, H., Nissimov, E., Pacheva, S.: Method of squared eigenfunction potentials in integrable hierarchies of KP type. Comm. Math. Phys. 193, 493–525 (1998)
Oevel, W., Carillo, S.: Squared eigenfunction symmetries for soliton equations: part I. J. Math. Anal. Appl. 217, 161–178 (1998)
Oevel, W., Carillo, S.: Squared eigenfunction symmetries for soliton equations: part II. J. Math. Anal. Appl. 217, 179–199 (1998)
Cheng, J.P., He, J.S., Hu, S.: The ghost symmetry of the BKP hierarchy. J. Math. Phys. 51, 053514 (2010)
Dickey, L.A.: On additional symmetries of the KP hierarchy and Sato’s backlund transformation. Commun. Math. Phys. 167, 227–233 (1995)
Acknowledgments
We are grateful to Prof. Folkert Mueller-Hoissen in Max-Planck-Institute for Dynamics and Self-Organization (Göttingen in Germany) for valuable discussions and suggestions. Chuanzhong Li is supported by the National Natural Science Foundation of China under Grant No. 11201251, the Zhejiang Provincial Natural Science Foundation under Grant Nos. LY15A010004, LY12A01007, the Natural Science Foundation of Ningbo under Grant Nos. 2015A610157, 2013A610105, 2014A610029. Maohua Li is supported by the Zhejiang Provincial Natural Science Foundation under Grant No. LY15A010005. Jingsong He is supported by the National Natural Science Foundation of China under Grant No. 11271210, K.C.Wong Magna Fund in Ningbo University. Jipeng Cheng and Kelei Tian are supported by the National Natural Science Foundation of China under Grant Nos. 11301526, 11201451.
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Communicated by G. Teschl.
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Li, C., Cheng, J., Tian, K. et al. Ghost symmetry of the discrete KP hierarchy . Monatsh Math 180, 815–832 (2016). https://doi.org/10.1007/s00605-015-0802-z
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DOI: https://doi.org/10.1007/s00605-015-0802-z