70 research outputs found
On the squared eigenfunction symmetry of the Toda lattice hierarchy
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.Comment: 17 pages, submitte
The applications of the gauge transformation for the BKP hierarchy
In this paper, we investigated four applications of the gauge transformation
for the BKP hierarchy. Firstly, it is found that the orbit of the gauge
transformation for the constrained BKP hierarchy defines a special -dimensional Toda lattice equation structure. Then the tau function of the
BKP hierarchy generated by the gauge transformation is showed to be the
Pfaffian. And the higher Fay-like identities for the BKP hierarchy is also
obtained through the gauge transformation. At last, the compatibility between
the additional symmetry and the gauge transformation of the BKP hierarchy is
proved.Comment: 19 pages, no figures. Submitte
The 2-component BKP Grassmanian and simple singularities of type D
It was proved in 2010 that the principal Kac--Wakimoto hierarchy of type
is a reduction of the 2-component BKP hierarchy. On the other hand, it is known
that the total descendant potential of a singularity of type is a
tau-function of the principal Kac--Wakimoto hierarchy. We find explicitly the
point in the Grassmanian of the 2-component BKP hierarchy (in the sense of
Shiota) that corresponds to the total descendant potential. We also prove that
the space of tau-functions of Gaussian type is parametrized by the base of the
miniversal unfolding of the simple singularity of type .Comment: 36 page
Squared Eigenfunction Symmetries for the BTL and CTL Hierarchies
In this paper, the squared eigenfunction symmetries for the BTL and CTL
hierarchies are explicitly constructed with the suitable modification of the
ones for the TL hierarchy, by considering the BTL and CTL constraints. Also the
connections with the corresponding additional symmetries are investigated: the
squared eigenfunction symmetry generated by the wave function can be viewed as
the generating function for the additional symmetries.Comment: 11pages, no figure
The "ghost" symmetry in the CKP hierarchy
In this paper, we systematically study the "ghost" symmetry in the CKP
hierarchy through its actions on the Lax operator, dressing operator,
eigenfunctions and the tau function. In this process, the spectral
representation of the eigenfunction is developed and the squared eigenfunction
potential is investigated.Comment: Accepted by J. Geom. Phy
The gauge transformation of the constrained semi-discrete KP hierarchy
In this paper, the gauge transformation of the constrained semi-discrete
KP(cdKP) hierarchy is constructed explicitly by the suitable choice of the
generating functions. Under the -step successive gauge transformation ,
we give the transformed (adjoint) eigenfunctions and the -function of the
transformed Lax operator of the cdKP hierarchy.Comment: 13 pages, accepted by Modern Physics Letters
The "Ghost" Symmetry of the BKP hierarchy
In this paper, we systematically develop the "ghost" symmetry of the BKP
hierarchy through its actions on the Lax operator , the eigenfunctions and
the function. In this process, the spectral representation of the
eigenfunctions and a new potential are introduced by using squared
eigenfunction potential(SEP) of the BKP hierarchy. Moreover, the bilinear
identity of the constrained BKP hierarchy and Adler-Shiota-van-Moerbeke formula
of the BKP hierarchy are re-derived compactly by means of the spectral
representation and "ghost" symmetry.Comment: 23pages, to appear in Journal of Mathematical Physic
The Recursion operators of the BKP hierarchy and the CKP Hierarchy
In this paper, under the constraints of the BKP(CKP) hierarchy, a crucial
observation is that the odd dynamical variable can be explicitly
expressed by the even dynamical variable in the Lax operator
through a new operator . Using operator , the essential differences
between the BKP hierarchy and the CKP hierarchy are given by the flow equations
and the recursion operators under the -reduction. The formal formulas
of the recursion operators for the BKP and CKP hierarchy under
-reduction are given. To illustrate this method, the two recursion
operators are constructed explicitly for the 3-reduction of the BKP and CKP
hierarchies. The flows of are generated from flows by the
above recursion operators, which are consistent with the corresponding flows
generated by the flow equations under 3-reduction.Comment: 19 page
Hirota Quadratic Equations for the Gromov--Witten Invariants of
Fano orbifold lines are classified by the Dynkin diagrams of type and
. It is known that the corresponding total descendant potential is a
tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that
in the A-case the Kac--Wakimoto hierarchies admit an extension and that the
total descendant potential is a tau-function of an extended Kac--Wakimoto
hierarchy. The goal of this paper is to prove that in the D-case the total
descendent potential is also a tau-function of an extended Kac--Wakimoto
hierarchy.Comment: 37 pages. Some typos and statements are corrected, and the title has
been change
Quantum torus symmetries of multicomponent modified KP hierarchy and reductions
In this paper, we construct the multicomponent modified KP hierarchy and its
additional symmetries. The additional symmetries constitute an interesting
multi-folds quantum torus type Lie algebra. By a reduction, we also construct
the constrained multicomponent modified KP hierarchy and its Virasoro type
additional symmetries.Comment: 11 Pages. arXiv admin note: text overlap with arXiv:1510.0895
- β¦