×
Geng, Lumin; Hu, Jianxun; Wu, Chao-Zhong

A KP-mKP hierarchy via pseudo-differential operators with two derivations. (English) Zbl 07892747

Nonlinearity 37, No. 9, Article ID 095016, 34 p. (2024).
Summary: By using pseudo-differential operators containing two derivations, we extend the Kadomtsev-Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker-Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.
{© 2024 IOP Publishing Ltd & London Mathematical Society}

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35S05 Pseudodifferential operators as generalizations of partial differential operators
14H70 Relationships between algebraic curves and integrable systems

Keywords:

Kadomtsev-Petviashvili hierarchy; tau function; Whitham hierarchy
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Alexandrov, A., Open intersection numbers, matrix models and MKP hierarchy, J. High Energy Phys., JHEP03(2015)042, 2015 · Zbl 1388.81463 · doi:10.1007/JHEP03(2015)042
[2] Buryak, A., Equivalence of the open KdV and the open Virasoro equations for the moduli space of Riemann surfaces with boundary, Lett. Math. Phys., 105, 1427-48, 2015 · Zbl 1323.35158 · doi:10.1007/s11005-015-0789-3
[3] Buryak, A., Open intersection numbers and the wave function of the KdV hierarchy, Mosc. Math. J., 16, 27-44, 2016 · Zbl 1339.35265 · doi:10.17323/1609-4514-2016-16-1-27-44
[4] Cheng, J.; Milanov, T., The 2-component BKP Grassmannian and simple singularities of type D, Int. Math. Res. Not., 2021, 17875-923, 2021 · Zbl 1487.37082 · doi:10.1093/imrn/rnz325
[5] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations: IV. A new hierarchy of soliton equations of KP-type, Physica D, 4, 343-65, 198182 · Zbl 0571.35100 · doi:10.1016/0167-2789(82)90041-0
[6] Date, E.; Kashiwara, M.; Jimbo, M.; Miwa, T., Transformation groups for soliton equations, pp 39-119, 1983, World Scientific · Zbl 0571.35098
[7] De Sole, A.; Kac, V. G.; Valeri, D., On Lax operators, Jpn. J. Math., 17, 63-116, 2022 · Zbl 1522.37077 · doi:10.1007/s11537-021-2134-1
[8] Dickey, L. A., Soliton Equations and Hamiltonian Systems, 2003, World Scientific · Zbl 1140.35012
[9] Geng, L.; Hu, J.; Wu, C-Z, On Lax equations of the two-component BKP hierarchy, Physica D, 449, 2023 · Zbl 1517.35274 · doi:10.1016/j.physd.2023.133748
[10] Hirota, R., The Direct Method in Soliton Theory, 2004, Cambridge University Press · Zbl 1099.35111
[11] Jimbo, M.; Miwa, T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19, 943-1001, 1983 · Zbl 0557.35091 · doi:10.2977/prims/1195182017
[12] Kontsevich, M., Intersection theory on the moduli space of curves and the matrix airy function, Commun. Math. Phys., 147, 1-23, 1992 · Zbl 0756.35081 · doi:10.1007/BF02099526
[13] Krichever, I. M., The τ-function of the universal Whitham hierarchy, matrix models and topological field theories, Commun. Pure Appl. Math., 47, 437-75, 1994 · Zbl 0811.58064 · doi:10.1002/cpa.3160470403
[14] Liu, S-Q; Wu, C-Z; Zhang, Y., On the Drinfeld-Sokolov hierarchies of D type, Int. Math. Res. Not., 2011, 1952-96, 2011 · Zbl 1221.35458 · doi:10.1093/imrn/rnq138
[15] Lu, J.; Wu, C-Z, Bilinear equation and additional symmetries for an extension of the Kadomtsev-Petviashvili hierarchy, Math. Phys. Anal. Geom., 24, 27, 2021 · Zbl 1487.37086 · doi:10.1007/s11040-021-09401-6
[16] Ma, S.; Wu, C-Z; Zuo, D., Infinite-dimensional Frobenius manifolds underlying an extension of the dispersionless Kadomtsev-Petviashvili hierarchy, J. Geom. Phys., 161, 2021 · Zbl 1477.53113 · doi:10.1016/j.geomphys.2020.104006
[17] Oevel, W.; Rogers, C., Gauge transformations and reciprocal links in 2+1 dimensions, Rev. Math. Phys., 5, 299-330, 1993 · Zbl 0780.35101 · doi:10.1142/S0129055X93000073
[18] Shiota, T., Prym varieties and soliton equations, pp 407-48, 1989, World Scientific · Zbl 0766.14020
[19] Szablikowski, B. M.; Blaszak, M., Dispersionful analog of the Whitham hierarchy, J. Math. Phys., 49, 2008 · Zbl 1152.81615 · doi:10.1063/1.2970774
[20] Witten, E., Two-Dimensional Gravity and Intersection Theory on the Moduli Space (Surveys in Differential Geometry), pp 243-310, 1990, International Press of Boston · Zbl 0757.53049
[21] Wu, C-Z; Zhou, X., An extension of the Kadomtsev-Petviashvili hierarchy and its hamiltonian structures, J. Geom. Phys., 106, 327-41, 2016 · Zbl 1347.37119 · doi:10.1016/j.geomphys.2016.04.008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.