On the squared eigenfunction symmetry of the Toda lattice hierarchy

Abstract

Abstract. The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively. Then after the Fay-like identities and some important relations about the wave functions are investigated, the action of the squared eigenfunction related to the additional symmetry on the tau function is derived, which is equivalent to the Adler-Shiota-van Moerbeke (ASvM) formulas.

Bilinear Identities and Squared Eigenfunction Symmetries of the BC r -KP Hierarchy

“…The squared eigenfunction symmetry can be traced back to [22], where Oevel studied the solutions of the constrained KP hierarchy in the first time. Then it is widely investigated in [4,5,13,14]. The squared eigenfunction symmetry can be used to define the new integrable system, such as the extended integrable system [17,18] and the symmetry constraint [6,7,[19][20][21][24][25][26][27] and the additional symmetry [1,2,10,28].…”

“…The squared eigenfunction symmetry can be used to define the new integrable system, such as the extended integrable system [17,18] and the symmetry constraint [6,7,[19][20][21][24][25][26][27] and the additional symmetry [1,2,10,28]. Recently, many researches have been done in the squared eigenfunction symmetries, for instance, the Toda lattice hierarchy and its sub hierarchy of B and C type [4,5], the discrete KP [14] and modified discrete KP [13] hierarchies are investigated recently. In this paper, we will consider some properties of the squared eigenfunction symmetries of the BC r -KP hierarchy and its constrained case.…”

The “ghost” symmetry in the CKP hierarchy

“…The "ghost" symmetry has attracted many researches recently. For example, recently the "ghost" symmetries for the BKP hierarchy [36], discrete KP hierarchy [37], the Toda lattice hierarchy [38] and its B type and C type cases [39] are all studied.…”

Squared Eigenfunction Symmetries for the BTL and CTL Hierarchies

“…The squared eigenfunction symmetry has many applications in the integrable system. For example, 1) symmetry constraint [5,7,[9][10][11][12][13][14] can be defined by identifying the squared eigenfunction symmetry with the usual flow of the integrable hierarchy; 2) the connection with the additional symmetry [8,[15][16][17], which is the symmetry depending explicitly on the space and time variables [18][19][20][21][22][23][24][25][26]; 3) the extended integrable systems [27,28], which contain the integrable equations with self-consistent sources, can be constructed with the help of the squared eigenfunction symmetry. Recently, the squared eigenfunction symmetries for the BKP hierarchy and the discrete KP hierarchy are systematically developed in [15] and [16] respectively.…”

“…Recently, the squared eigenfunction symmetries for the BKP hierarchy and the discrete KP hierarchy are systematically developed in [15] and [16] respectively. Also the squared eigenfunction symmetry for the TL hierarchy and its connection with the additional symmetry are investigated in [17]. In this paper, we will concentrate on the construction of the squared eigenfunction symmetry of the BTL and CTL hierarchies.…”