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We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov-Witten total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities., 25 pages

We introduce a single tau function that represents the CKP hierarchy into a generalized Hirota "bilinear" equation. The actions on the tau function by additional symmetries for the hierarchy are calculated, which involve strictly more than a central extension of the $w^C_\infty$-algebra. As an application, for Drinfeld-Sokolov hierarchies of type C that are equivalent to certain reductions of the CKP hierarchy, their Virasoro symmetries are proved to be non-linearizable when acting on the tau function., 29 pages, published version

We consider Ramanujan functions and a family of functions whose properties are close to those of Ramanujan functions. We prove the algebraic independence of these functions over the field of rational functions with complex coefficients. We also prove a bound for the multiplicity of the zeros of polynomials in these functions.

We prove that for every integer the Diophantine equation , where is the Ramanujan -function, has a solution in positive integers satisfying the condition . We also consider similar questions in residue fields modulo a large prime .

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series., Comment: 32 pages, corrected typos

We discuss the notion of the Bergman tau-function on Hurwitz spaces, spaces of abelian and quadratic differentials on Riemann surfaces. The Bergman tau-function on Hurwitz spaces is nothing but isomonodromic tau-function of Hurwitz Frobenius manifolds; it also appears in genus one contribution to partition function of Hermitian two-matrix model. The Bergman tau-function on spaces of abelian and quadratic differentials appears in the problem of holomorphic factorization of determinants of Laplacians in flat metrics with conical singularities over Riemann surfaces.

We study the dependence of the tau function of Painlev\'e I equation on the generalized monodromy of the associated linear problem. In particular, we compute connection constants relating the tau function asymptotics on five canonical rays at infinity. The result is expressed in terms of dilogarithms of cluster type coordinates on the space of Stokes data., Comment: 13 pages, 2 figures

In this paper, we have considered the q deformation of the AKNS-D hierarchy, proved the bilinear identity and obtained the τ function of the q deformed AKNS-D hierarchy.

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of 'shift symmetries'. They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can also be characterized by a factorization problem of [Z x Z] matrices. The associated Lax operators turn out to be quotients of first-order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz–Ladik (equivalently, the relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

We study the KP hierarchy through its relationship with S-functions. Using results from the classical theory of symmetric functions, the Plucker equations for the hierarchy are derived from the tau function bilinear identity and are given in terms of composite S-functions. Their connection to the Hirota bilinear form of the hierarchy is clarified. A novel combinatorial proof is given of the fact that Schur polynomials solve the KP hierarchy. We show how the analysis can be carried through for the BKP hierarchy in a completely parallel fashion, with the S-functions replaced by Schur Q-functions.

From the perspective of Hirota's bilinear difference equation, the connection of the tau function and the conformal symmetry is shown to become apparent and the Virasoro constraint is naturally derived. In this formalism, the compatibility of the flow provided by conformal symmetry and the KP flow is manifest.

In this paper, we defined a new multi-component BKP hierarchy which takes values in a commutative subalgebra of $gl(N,\mathbb C)$. After this, we give the gauge transformation of this commutative multi-component BKP (CMBKP) hierarchy. Meanwhile we construct a new constrained CMBKP hierarchy which contains some new integrable systems including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed., 15 pages

In this paper, we will present some of our results on the soliton hierarchy with self–consistent sources (SHSCSs). The Kadomtsev–Petviashvili (KP) hierarchy will be used as an illustrative example to show the method to construct the SHSCSs. Some properties of the KP hierarchy with self–consistent sources will also be given, such as the dressing approach, the Wronskian solutions (including soliton solutions), its bilinear identities and the tau function.

We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this proves a conjecture of Milanov-Tseng that relates the total descendent potential of the orbifold $C_{k,m}$ with a tau function of the bigraded Toda hierarchy., Comment: 17 pages

We compare and contrast three different methods for the construction of the differential relations satisfied by the fundamental Abelian functions associated with an algebraic curve. We realize these Abelian functions as logarithmic derivatives of the associated sigma function. In two of the methods, the use of the tau function, expressed in terms of the sigma function, is central to the construction of differential relations between the Abelian functions., Comment: 25 pages

The two-dimensional Toda lattice (2DTL) is a well-known (2+1)-dimensional integrable system that admits a large class of line-soliton solutions. We classify these soliton solutions by exploiting the structure of the Casoratian expression for its tau function. In general, these solutions consist of unequal numbers of 'incoming' and 'outgoing' line solitons. We classify the incoming and outgoing line solitons based on asymptotic analysis of the tau function of the 2DTL as the discrete variable tends to infinity. We also identify various subclasses of solutions and characterize some of them in terms of the amplitudes and directions of the interacting solitons. As a special case, we obtain the reduction in the soliton solutions of the one-dimensional Toda lattice. Throughout, we point out the similarities with—and the differences from—the corresponding results for the Kadomtsev–Petviashvili equation.

Two kinds of symmetries, Sato's Backlund transformations and additional symmetries, for the discrete KP (dKP) hierarchy are introduced, and the ASvM formula which demonstrates the equivalence of these two kinds of symmetries is obtained. In this process the Fay identity and the difference Fay identity of the dKP hierarchy are introduced and the ASvM formula in the form of tau function is calculated.

We have classified a tau function for the hypergeometric solutions of the Painleve VI equation constructed by Shah and Woodhouse (2006 J. Phys. A: Math. Gen. 39, 12265–9) through twistor methods. We have shown that the tau function is the product of a Toeplitz determinant and a power of the time variable t. In a suitable trivialization of the twistor bundle, the symbol of this Toeplitz determinant is the minus of the off-diagonal entry in the patching matrix. The method can also be applied to other solutions obtained from the Ward ansatze.

The tau function, introduced by the 'Kyoto School' as a central element in the description of soliton equation hierarchies, is identified with the determinant of a family of linear operators solving linear, constant-coefficient PDE in the hierarchy variables, for the Kadomtsev-Petviashvili (KP) hierarchy. For Gel'fand-Levitan-Marchenko integral operators, the tau function is the Fredholm determinant; the author gives a new proof of this fact. He shows further that the determinant of a suitably chosen family of finite-dimensional matrices is a tau function which gives rise to rational solutions of the KP equation. Finally, he proves the 'vertex operator identity' for tau functions in the Fredholm determinant case.