Constrained discrete KP hierarchy: the constraint on the tau functions and gauge transformations

Jia-Qi Song   ; Chong Li   ; Ji-Peng Cheng   ; Min-Ru Chen

Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 3, pp. 387-409

Résumé

We investigate two topics of the constrained discrete KP hierarchy: the constraint on the tau functions and gauge transformations. For the constrained dKP hierarchy, the constraint on the Lax operator $(L^k)_{<0}=q\Delta^{-1}r$ is shown to be equivalent to a constraint on the tau function, which provides another description of the constrained dKP hierarchy. We also find that there are two different choices of the generating functions in gauge transformation operators of the constrained discrete KP hierarchy.

Détail

Keywords: discrete KP hierarchy, constrained discrete KP hierarchy, gauge transformation, tau function.

Bibliographie

[1] B. A. Kupershmidt, Discrete Lax Equations and Differential-Difference Calculus, Astérisque, 123, Soc. Math. France, Paris, 1985 | MR

[2] L. Haine, P. Iliev, “Commutative rings of difference operators and an adelic flag manifold”, Internat. Math. Res. Notices, 2000:6 (2000), 281–323 | DOI | MR | Zbl

[3] S. Liu, Y. Cheng, “Sato's Bäcklund transformation, additional symmtries and ASvM formular for the discrete KP hierarchy”, J. Phys. A: Math. Theor., 43:13 (2010), 135202, 17 pp. | DOI | MR

[4] X. Wang, J. Zhu, Z. Qiao, “New solutions to the differential-difference KP equation”, Appl. Math. Lett., 113 (2021), 106836 | DOI | MR

[5] J. Cheng, J. He, “Miura and auto-Bäcklund transformations for the discrete KP and mKP hierarchies and their constrained cases”, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 187–197 | DOI | MR

[6] S. Jian, J. Cheng, “The squared eigenfunction symmetries of the discrete KP and modified discrete KP hierarchies under the Miura transformations”, Modern Phys. Lett. B, 32:27 (2018), 1850326, 20 pp. | DOI | MR

[7] S. W. Liu, Y. Cheng, J. S. He, “The determinant representation of the gauge transformation for the discrete KP hierarchy”, Sci. China Math., 53:5 (2010), 1195–1206 | DOI | MR

[8] C. Li, J. Cheng, K. Tian, M. Li, J. He, “Ghost symmetry of the discrete KP hierarchy”, Monatsh. Math., 180:4 (2016), 815–832 | DOI | MR

[9] X.-L. Sun, D.-J. Zhang, X.-Y. Zhu, D.-Y. Chen, “Symmetries and Lie algebra of the differential-difference Kadomstev–Petviashvili hierarchy”, Modern Phys. Lett. B, 24:10 (2010), 1033–1042 | DOI | MR

[10] M. Adler, P. van Moerbeke, “Vertex operator solutions to the discrete KP-hierarchy”, Commun. Math. Phys., 203:1 (1999), 185–210, arXiv: solv-int/9912014 | DOI | MR

[11] L. A. Dickey, “Modified KP and discrete KP”, Lett. Math. Phys., 48:3 (1999), 277–289 | DOI | MR

[12] H. Aratyn, E. Nissimov, S. Pacheva, “Method of squared eigenfunction potentials in integrable hierarchies of KP type”, Commun. Math. Phys., 193:3 (1998), 493–525 | DOI | MR | Zbl

[13] M. J. Ablowitz, B. Prinari, A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, London Mathematical Society Lecture Note Series, 302, Cambridge Univ. Press, Cambridge, 2004 | MR

[14] M. Li, C. Li, K. Tian, J. He, Y. Cheng, “Virasoro type algebraic structure hidden in the constrained discrete Kadomtsev–Petviashvili hierarchy”, J. Math. Phys., 54:4 (2013), 043512, 11 pp. | DOI | MR

[15] J. Cheng, J. He, “The full Virasoro symmetry for the constrained discrete KP hierarchy”, Chinese Ann. Math. Ser. A, 41:1 (2020), 17–38 (in Chinese) | MR

[16] L.-L. Chau, J. C. Shaw, H. C. Yen, “Solving the KP hierarchy by gauge transformations”, Commun. Math. Phys., 149:2 (1992), 263–278 | DOI | MR

[17] L.-L. Chau, J.-C. Shaw, M.-H. Tu, “Solving the constrained KP hierarchy by gauge transformations”, J. Math. Phys., 38:8 (1997), 4128–4137 | DOI | MR

[18] H. Aratyn, E. Nissimov, S. Pacheva, “Constrained KP hierarchies: additional symmetries, Darboux–Bäcklund solutions and relations to multi-matrix models”, Internat. J. Modern Phys. A, 12:7 (1997), 1265–1340 | DOI | MR

[19] J. He, Y. Li, Y. Cheng, “Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy”, J. Math. Phys., 44:9 (2003), 3928–3960 | DOI | MR

[20] W. Oevel, “Darboux theorems and Wronskian formulas for integrable system I: constrained KP flows”, Phys. A, 195:3–4 (1993), 533–576 | DOI | MR

[21] W. Oevel, “Darboux transformations for integrable lattice systems”, Nonlinear Physics: Theory and Experiment, Proceedings of the First Workshop on Nonlinear Physics, Theory and Experiment: Nature, Structure and Properties of Nonlinear Phenomena (Le Siernuse, Gallipoli (Lecce), Italy, June 29 –July 7, 1995), eds. E. Alfinito, M. Boiti, L. Martina, F. Pempinelli, World Sci., Singapore, 1996, 233–240 | MR

[22] M. Li, J. Cheng, J. He, “The gauge transformation of the constrained semi-discrete KP hierarchy”, Modern Phys. Lett. B, 27:6 (2013), 1350043, 13 pp. | DOI | MR

[23] A. Kundu, W. Strampp, W. Oevel, “Gauge transformations of constrained KP flows: new integrable hierarchies”, J. Math. Phys., 36:6 (1995), 2972–2984 | DOI | MR