KP和mKP可积系列的平方本征对称和Miura变换

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 10-19.

KP和mKP可积系列的平方本征对称和Miura变换

耿露敏,陈慧展,李娜,程纪鹏*

^{中国矿业大学数学学院江苏徐州 221116}

收稿日期:2018-09-27 出版日期:2020-01-26 发布日期:2020-04-08
通讯作者: 程纪鹏
基金资助:
中国博士后科学基金(2016M591949);江苏省博士后科学基金(1601213C)

The Squared Eigenfunction Symmetries and Miura Transformations for the KP and mKP Hierarchies

Lumin Geng,Huizhan Chen,Na Li,Jipeng Cheng*

^{School of Mathematics, China University of Mining Technology, Jiangsu Xuzhou 221116}

Received:2018-09-27 Online:2020-01-26 Published:2020-04-08
Contact: Jipeng Cheng
Supported by:
中国博士后科学基金(2016M591949);江苏省博士后科学基金(1601213C)

摘要/Abstract

摘要：

该文讨论了KP和mKP可积系列及其约束情形的平方本征对称与Miura变换和反-Miura变换的关系.

关键词: KP和mKP可积系列, 约束KP和约束mKP可积系列, 平方本征对称, Miura变换和反-Miura变换

Abstract:

In this paper, we discuss the relations of the squared eigenfunction symmetry and the Miura and auti-Miura transformations for the KP and mKP hierarchies and their constrained cases.

Key words: KP and mKP hierarchies, Constrained KP and constrained mKP hierarchies, Squared eigenfunction symmetry, Miura and auti-Miura transformations

中图分类号:

O411.1

耿露敏,陈慧展,李娜,程纪鹏. KP和mKP可积系列的平方本征对称和Miura变换[J]. 数学物理学报, 2020, 40(1): 10-19.

Lumin Geng,Huizhan Chen,Na Li,Jipeng Cheng. The Squared Eigenfunction Symmetries and Miura Transformations for the KP and mKP Hierarchies[J]. Acta mathematica scientia,Series A, 2020, 40(1): 10-19.

参考文献 33

1	Date E, Jimbo M, Kashiwara M, Miwa T. Transformation Groups for Soliton Equations//Stone M. Nonlinear Inregrable Systems-Classical and Quantum Theory. Singapore:World Scientific, 1983:427-507
2	Dickey L A. Soliton Equations and Hamiltonian Systems. Singapore:World Scientific, 2003
3	Cheng Y , Li Y S . The constraint of the KP equation and its special solutions. Phys Lett A, 1991, 157, 22- 26
4	Konopelchenko B , Sidorenko J , Strampp W . (1+1)-Dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems. Phys Lett A, 1999, 157, 17- 21
5	Cheng Y . Constraints of the KP hierarchy. J Math Phys, 1992, 33, 3774- 3782
6	Cheng J P . The integral type gauge transformation and the additional symmetry for the constrained KP hierarchy. Acta Math Sci, 2015, 35B, 1111- 1121
7	Gao X , Li C Z , He J S . Recursion relations for the constrained multi-component KP hierarchy. Acta Math Sin, 2017, 33, 1578- 1586
8	Aratyn H , Nissimov E , Pacheva S . Virasoro symmetry of constrained KP hierarchies. Phys Lett A, 1997, 288, 164- 175
9	Sidorenko J , Strampp W . Symmetry constraints of the KP hierarchy. Inverse Problems, 1991, 7, L37- L43
10	Kashiwara M , Miwa T . The τ function of the Kadomtsev-Petviashvili equation transfromation groups for soliton equations, I. Proc Japan Acad A, 1981, 57, 342- 347
11	Jimbo M , Miwa T . Solitons and infinite dimensional Lie algebras. Publ RIMS Kyoto Univ, 1983, 19, 943- 1001
12	Oevel W , Rogers C . Gauge transformations and reciprocal links in 2+1 dimensions. Rev Math Phys, 1993, 5, 299- 330
13	Kupershmidt B A . Mathematics of dispersive water waves. Commun Math Phys, 1985, 99, 51- 73
14	Kiso K . A remark on the commuting flows defined by Lax equations. Prog Theo Phys, 1990, 83, 1108- 1125
15	Shaw J C , Tu M H . Miura and auto-Backlund transformations for the cKP and cmKP hiearchies. J Math phys, 1997, 38, 5756- 5773
16	Shaw J C , Yen Y C . Miura and Backlund transformations for hierarchies of integrable equations. Chin J Phys, 1993, 31, 709- 719
17	Cheng J P , Li M H , Tian K L . On the modified KP hierarchy:tau functions, squared eigenfunction symmetries and additional symmetries. J Geom Phys, 2018, 134, 19- 37
18	Cheng J P . The gauge transformation of the modified KP hierarchy. J Nonlin Math Phys, 2018, 25, 66- 85
19	Oevel W , Carillo S . Squared eigenfunction symmetries for soliton equations:Part I. J Math Anal Appl, 1998, 217, 161- 178
20	Oevel W , Carillo S . Squared eigenfunction symmetries for soliton equations:Part II. J Math Anal Appl, 1998, 217, 179- 199
21	Oevel W . Darboux theorems and Wronskian formulas for intergrable system I:Constrained KP flows. Phys A, 1993, 195, 533- 576
22	Aratyn H , Nissimov E , Pacheva S . Method of squared eigenfunction potentials in integrable hierarchies of KP type. Commun Math Phys, 1998, 193, 493- 525
23	Adler M , Shiota T , Van Moerbeke P . A Lax representation for the vertex operator and the central extension. Commun Math Phys, 1995, 171, 547- 588
24	Dickey L A . On additional symmetries of the KP hierarchy and Sato's Backlund transformation. Commun Math Phys, 1995, 167, 227- 233
25	Liu X J , Zeng Y B , Lin R L . A new extended KP hierarchy. Phys Lett A, 2008, 372, 3819- 3823
26	Liu X J , Zeng Y B , Lin R L . An extended two-dimensional Toda lattice hierarchy and two-dimensional Toda lattice with self-consistent sources. J Math Phys, 2008, 49, 093506
27	Cheng J P , He J S , Hu S . The "ghost" symmetry the BKP hierarchy. J Math Phys, 2010, 51, 053514
28	Cheng J P , He J S . On the squared eigenfunction symmetry of the Toda lattice hierarchy. J Math Phys, 2013, 54, 023511
29	Cheng J P , He J S . Squared eigenfunction symmetries for the BTL and CTL hierarchies. Commun Theor Phys, 2013, 59, 131- 136
30	Miura R M . Korteweg-de Vries equation and generalizations I:A remarkable explicit nonlinear transformation. J Math Phys, 1968, 9, 1202- 1204
31	Cheng J P . Miura and auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies. J Nonlin Math Phys, 2017, 1, 7- 19
32	Konopelchenko B G . On the gauge-invariant description of the evolution equations integrable by Gelfand-Dikij spectral problems. Phys Lett A, 1982, 92, 323- 327
33	Kupershmidt B A . Canonical property of the Miura maps between the MKP and KP hierarchies, Continuous and Discrete. Commun Math Phys, 1995, 167, 351- 371

[1]	王晓翊,程纪鹏,李红云. 修正离散KP系列的流方程[J]. 数学物理学报, 2019, 39(2): 386-392.
[2]	韩祥临,汪维刚,莫嘉琪. 流行性病毒传播生态动力学系统[J]. 数学物理学报, 2019, 39(1): 200-208.
[3]	相培, 傅景礼. 二次曲面区域泊松方程第一边值问题的格林函数解法[J]. 数学物理学报, 2018, 38(4): 800-809.
[4]	孙云, 吴建光, 王东, 唐绪兵. 关于厄米特多项式的新微分公式及其在量子光学中的应用[J]. 数学物理学报, 2016, 36(4): 795-808.