Ghost symmetry of the discrete KP hierarchy

In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction (adjoint eigenfunction) and the spectral representation of its Baker–Akhiezer function and adjoint Baker–Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.     

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 (2+1)$(2+1)$-dimensional Toda lattice equation structure. Then the tau function of the BKP hierarchy generated by the gauge transformation is shown 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 proven.     

Extensions of the discrete KP hierarchy and its strict version

Theoretical and Mathematical Physics - We show that both the dKP hierarchy and its strict version can be extended to a wider class of deformations satisfying a larger set of Lax equations. We prove...     

The Lax representation and Darboux transformation for constrained flows of the AKNS hierarchy

By using a general scheme for decomposing a zero-curvature equation into two commutingx- andt _n -finite-dimensional integrable Hamiltonian systems (FDIHS), a systematic deduction of the Lax representation for all constrained flows of the AKNS hierarchy from the adjoint representation of the two auxiliary linear problems is presented. The Darboux transformation for these FDIHSs is derived.Supported by the Chinese National Basic Research Project Nonlinear Science     

KP hierarchy for Hodge integrals

Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten's conjecture, Virasoro constrains, Faber's λg-conjecture, etc. Among other results we show that a properly arranged generating function for Hodge integrals satisfies the equations of the KP hierarchy.     

Darboux transformations for the strict KP hierarchy

Theoretical and Mathematical Physics - We introduce the notion of Darboux transformations for the strict KP hierarchy. We previously showed that solutions of this integrable hierarchy can be...     

Reductions of the strict KP hierarchy

Theoretical and Mathematical Physics - Let $$R$$ be a commutative complex algebra and $$ \partial $$ be a $$ \mathbb{C} $$ -linear derivation of $$R$$ such that all powers of $$ \partial $$ are...     

Recursion relations for the constrained multi-component KP hierarchy

In this paper, we define a new constrained multi-component KP(cMKP) hierarchy which contains the constrained KP(cKP) hierarchy as a special case. We derive the recursion operator of the constrained multi-component KP hierarchy. As a special example, we also give the recursion operator of the constrained two-component KP hierarchy.     

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions arise in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.     

Group varieties related to the KP hierarchy

Some remarkable relations between group varieties and the solutions of a physically important class of differential equations, called KP hierarchy, are found. In particular, it is proved that to each solution in a certain class, including a lot of physically important solutions such as the famous n-solitons, there is associated in a natural way a group variety.     

The KP hierarchy, branched covers, and triangulations

The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plücker relations from geometry. We give a solution to the Plücker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the m-hypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond.     

Constraints of the KP hierarchy and the bilinear method

We consider generalizedk-constraints of the KP hierarchy where the Lax operatorL is forced to satisfy L _– ^k =q^–1r. We study the effect of those constraints on the bilinear equations.     

Connections on the total Picard sheaf and the KP hierarchy

A family of connections is introduced on a direct limit of Poincaré sheaves of a curve. These connections descend to connections on the corresponding direct limit of Picard sheaves, defined globally on the Jacobian of the curve. All such connections are classified; in particular they are all flat. Krichever's algebro-geometric solutions of the KP equations are recovered upon trivializing the resulting \tdD-modules over the complement of the divisor.Supported in part by NSF grant no. 58-1353149.     

A pfaffian version of the modified discrete KP equation

We first present Casorati determinant and Gram-type determinant solutions to a modified discrete KP equation and then produce a pfaffianized version of modified discrete KP equations by using Hirota and Ohta's pfaffianization procedure. The solutions to a coupled modified discrete KP equation are expressed by two types of pfaffians.     

Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy

The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.     

Quasiperiodic solutions of the discrete Chen-Lee-Liu hierarchy

Using the Lax matrix and elliptic variables, we decompose the discrete Chen-Lee-Liu hierarchy into solvable ordinary differential equations. Based on the theory of the algebraic curve, we straighten the continuous and discrete flows related to the discrete Chen-Lee-Liu hierarchy in Abel-Jacobi coordinates. We introduce the meromorphic function ?, Baker-Akhiezer vector \(\bar \psi \) , and hyperelliptic curve ?_N according to whose asymptotic properties and the algebro-geometric characters we construct quasiperiodic solutions of the discrete Chen-Lee-Liu hierarchy.