On a Symplectic Bigraded Toda Hierarchy

In this paper, we construct a symplectic bigraded Toda hierarchy which contains an symplectic deformation of the original Toda lattice hierarchy. In particular, we give the rational solutions which are expressed by the products of the symplectic Schur polynomials.

Regular Solution and Lattice Miura Transformation of Bigraded Toda Hierarchy

The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are given.

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper,the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations.After this,two kinds of Darboux transforma-tions in different directions and their mixed Darboux transformations of this hierarchy are constructed.These symmetries and Darboux transformations might be useful in Gromov-Witten theory of CP1.

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.

A NEW DISCRETE INTEGRABLE COUPLING SYSTEM AND ITS HAMILTONIAN STRUCTURE FOR THE MODIFIED TODA LATTICE HIERARCHY

We present a new discrete integrable coupling system by using the matrix Lax pair U, V C s/（4）. A novel spectral problem of modified Toda lattice soliton hierarchy is considered. Then, a new discrete integrable coupling equation hierarchy is obtained through the method of the enlarged Lax pair. Finally, we obtain the Hamiltonian structure of the integrable coupling system of the soliton equation hierarchy using the matrix-form trace identity. This discrete integrable coupling system includes a kind of a modified Toda lattice hierarchy.

GUE via Frobenius Manifolds.I.From Matrix Gravity to Topological Gravity and Back

Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov-Witten invariants of the complex projective line.In this paper,we give a direct proof of Dubrovin’s result.We also present in a diagram the recent progress on topological gravity and matrix gravity.

On a Reduction of the Kadomtsev-Petviashvili Hierarchy

This paper analyzes the reduction of the well known Kadomtsev-Petviashvili hierarchy. The reduction yields a previously unknown dispersion counterpart of the dispersionless hierarchy which has a Lax function of the form p＋u（x）（p-φ）^-1＋v（x）（p-φ）^-2. This paper also describes the bihamiltonian structure of the reduced hierarchy using Dirac reduction and proves that the approximation for the reduced hierarchy up to the second order of the dispersion parameter coincides with the hierarchy of integrable systems constructed from a particular twodimensional Frobenius manifold using the approach of Dubrovin and Zhang.