A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

We consider the 2-dimensional Toda lattice tau functions τ_n(t,s;η,θ) deforming the probabilities τ_n(η,θ) that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc (η,θ) of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painlevé VI equation.     

Punctured plane partitions and the q-deformed Knizhnik-Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik-Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley-Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ²-weighted punctured cyclically symmetric transpose complement plane partitions where τ=−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ²-enumerations of vertically symmetric alternating sign matrices and modifications thereof.     

On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller?s seminal paper. In particular, this paper extends Feller?s results for continuous Q-functions to measurable Q-functions and provides additional results.     

Differential Algebras with Banach-Algebra Coefficients II: The Operator Cross-Ratio Tau-Function and the Schwarzian Derivative

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules and then consider the classical Baker and τ-functions as operator-valued. Following from Part I we produce a pre-determinant structure for a class of τ-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map leads to an operator τ-function. We extend to this setting the operator cross-ratio which had previously been used to produce the scalar-valued τ-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes of a given projection. We link directly this cross-ratio with Fay’s trisecant identity for the τ-function. By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of an operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall–Chaundy C*-algebra (Part I) provides a link to the study of the KP hierarchy.     

Fermionic approach to soliton equations

In this paper we exploit the algebraic structure of the soliton equations and find solutions in terms of fermion particles. We show how determinants arise naturally in the fermionic approach to soliton equations. We write the τ-function for charged free fermions in terms of determinants. Examples of how to get soliton, rational and dromion solutions from τ-functions for the various soliton equations are given.     

A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.     

Solution of soliton equations in terms of neutral fermion particles

In this paper we exploit the algebraic structure of the soliton equations and find solutions in terms of neutral free fermion particles. We show how pfaffians arise naturally in the fermionic approach to soliton equations. We write the τ-function for neutral free fermions in terms of pfaffians. Examples of how to get soliton, rational and dromion solutions from τ-functions for the various soliton equations are given.     

Limit relation between toda chains and the elliptic SL(N, ?) top

We study a limit relation between the elliptic SL(N,?) top and Toda chains. We show that in the case of the nonautonomous SL(2, ?) top, whose equations of motion are related to the Painlevé VI equation, it turns out to be possible to modify the previously proposed procedure and in the limit obtain the nonautonomous Toda chain, whose equations of motion are equivalent to a particular case of the Painlevé III equation. We obtain the limit of the Lax pair for the elliptic SL(2, ?) top, which allows representing the equations of motion of the nonautonomous Toda chain as the equation for isomonodromic deformations.     

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.     

Tropical representation of Weyl groups associated with certain rational varieties

Starting from certain rational varieties blown-up from N(P¹), we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo-isomorphisms of the varieties. We develop an algebro-geometric framework of τ-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painlevé equations and its algebraic degree grows quadratically.     

Eventual periodicity of forced oscillations of the Korteweg-de Vries type equation

In this paper, we study the eventual periodicity of the initial boundary value problem (IBVP) for Korteweg-de Vries equation posed on a bounded domain. We show that if the boundary forcing is periodic of period τ, then the solution u of the IBVP at each spatial point becomes eventually time-periodic of period τ. In order to exhibit eventual periodicity, we approximate the solution of the IBVP using the Adomian decomposition method. We compare our work with the approximate solution of IBVP obtained by the homotopy perturbation method and present numerical experiments using Mathematica.     

Wave functions of the Toda chain with boundary interaction

We give an integral representation of the wave functions of the quantum N-particle Toda chain with boundary interaction. In the case of the Toda chain with a one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of -functions. The wave function of the Toda chain with a two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits a separation of variables. The separated difference equations coincide with the Baxter equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 346–364, February, 2005.     

Differential Algebras with Banach-Algebra Coefficients I: from C*-Algebras to the K-Theory of the Spectral Curve

We present an operator-coefficient version of Sato’s infinite-dimensional Grassmann manifold, and τ-function. In this setting the classical Burchnall–Chaundy ring of commuting differential operators can be shown to determine a C*-algebra. For this C*-algebra topological invariants of the spectral ring become readily available, and further, the Brown–Douglas–Fillmore theory of extensions can be applied. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall–Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the Burchnall–Chaundy C*-algebra extension by the compact operators provides a family of operator τ-functions.     

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a _n ,b _n )^T={f(?)} _n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.     

Long-time behavior of the difference solutions to the 2D GKS equation

We study the long-time behavior of the finite difference solution to the generalized Kuramoto-Sivashinsky equation in two space dimensions with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system and the upper semicontinuity d(A_h,τ,A)→0. Finally, we obtain the long-time stability and convergence of the difference scheme. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.     

Maximum principles for a fourth order equation from thin plate theory

This paper focuses on a nonlinear equation from thin plate theory of the form Δ(D(x)Δw)−(1−ν)[D,w]+c(x)f(w)=0. We obtain maximum principles for certain functions defined on the solution of this equation using P-functions or auxiliary functions of the types used by Payne [L.E. Payne, Some remarks on maximum principles, J. Anal. Math. 30 (1976) 421-433] and Schaefer [P.W. Schaefer, Solution, gradient, and laplacian bounds in some nonlinear fourth order elliptic equations, SIAM J. Math. Anal. 18 (1987) 430-434].     

On the Structure of the Set of E-Functions Satisfying Linear Differential Equations of Second Order

We prove a special case of Siegel's conjecture concerning the representability of E-functions in the form of polynomials in hypergeometric functions. We prove several assertions (formulated earlier by A. B. Shidlovskii) about the transcendence and linear independence of values of E-functions.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

L-Functions of Jacobi forms with Shimura type

For every Jacobi form of Shimura type over H × ℂ, a system of L-functions associated to it is given. These L-functions can be analytically continued to the whole complex plane and satisfy a kind of functional equation. As a consequence, Hecke’s inverse theorem on modular forms is extended to the context of Jacobi forms with Shimura type.     

Analysis of a two-phase field model for the solidification of an alloy

In this paper we present some theoretical results for a system of nonlinear partial differential equations that provide a phase field model for the solidification/melting of a metallic alloy. It is assumed that two different kinds of crystallization are possible. Consequently, the unknowns are the temperature τ and the phase field functions u and v. The time derivatives ut and vt appear in the equation for τ (the heat equation). On the other hand, the equations for u and v contain nonlinear terms where we find τ.