Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy∗ Chuanzhong LI1
Abstract In this paper, the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations. After this, two kinds of Darboux transformations in different directions and their mixed Darboux transformations of this hierarchy are constructed. These symmetries and Darboux transformations might be useful in GromovWitten theory of CP 1 . Keywords Extended Toda hierarchy, Ghost symmetry, Spectral representations, Hirota quadratic equation, Darboux transformation 2010 MR Subject Classification 37K05, 37K10, 37K20
1 Introduction The Toda lattice hierarchy (see [1]) as a completely integrable system has many important applications in mathematics and physics. Toda systems have many kinds of reductions or extensions, such as the extended Toda hierarchy (ETH for short) (see [2–4]), bigraded Toda hierarchy (BTH for short) (see [5–9]), extended multi-component Toda hierarchy (see [10]), extended ZN -Toda hierarchy (see [11]) and so on. With additional logarithm flows, the Toda lattice hierarchy becomes the extended Toda hierarchy (see [2]) which governs the Gromov-Witten invariant of CP 1 . That means the GromovWitten potential τ of CP 1 satisfies the Hirota quadratic equations (see [4]) of the ETH. The extended bigraded Toda hierarchy (EBTH for short) (see [5]) is an extension of the ETH. The Hirota bilinear equation of the EBTH was equivalently constructed in [6, 12]. Meanwhile it was proved to govern Gromov-Witten invariants of the total descendent potential of CN,M orbifolds (see [12]). The systematical studies on symmetries on lattice equations can be seen in [13]. As one kind of symmetries depending on time variables explicitly, the ghost symmetry was discovered by Oevel [14]. After that, it attracts a lot of research (see [15–22]). Aratyn used the method of squared eigenfunction potentials to construct the ghost symmetry of the KP hierarchy and connect this kind of symmetry with constrained KP hierarchy (see [17–19]). One S function Manuscript received June 28, 2018. Revised November 4, 2018. of Mathematics and Statistics, Ningbo University, Ningbo 315211, Zhejiang, China; College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China. E-mail: [email protected] ∗ This work was supported by the National Natural Science Foundation of China (No. 11571192) and K. C. Wong Magna Fund in Ningbo University. 1 School
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was used to represent (2 + 1)-dimensional hierarchies of the KP equation, the modified KP equation and the Dym equation (see [20–21]. Our group gave a good construction of the ghost symmetry of the discrete KP hierarchy (see [23]) and the BKP hierarchy (see [22]). Among many analytical methods, the Darboux transformation is one of the efficient methods to generate the soliton solutions for
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