Lattice Hierarchy Research Articles

Two-Dimensional Dispersionless Toda Lattice Hierarchy: Symmetry, New Extension, Hodograph Solutions, and Reduction

The symmetry for two-dimensional (2D) dispersionless Toda lattice hierarchy (dTLH) is firstly derived, and then the 2D dTLH is extended based on the symmetry constraint. The commutativity of two different flows for this new hierarchy is shown, which leads to the 2D dToda lattice equation with self-consistent sources (dTLESCSs) together with its conservation equation. The hodograph solutions to 2D dTLESCSs are also given. One dimensional reduction of extended 2D dTLH is finally investigated by finding the constraint, and a one-dimensional dTLESCS is shown.

Skew-Orthogonal Polynomials and Pfaff Lattice Hierarchy Associated With an Elliptic Curve

Abstract Starting with a skew-symmetric inner product over an elliptic curve, we propose the concept of elliptic skew-orthogonal polynomials. Inspired by the Landau–Lifshitz hierarchy and its corresponding time evolutions, we obtain the recurrence relation and the $\tau $-function representation for such a novel class of skew-orthogonal polynomials. Furthermore, a bilinear integral identity is derived through the so-called Cauchy–Stieljes transformation, from which we successfully establish the connection between the elliptic skew-orthogonal polynomials and an elliptic extension of the Pfaff lattice hierarchy.

From wave functions to tau-functions for the Volterra lattice hierarchy

From wave functions to tau-functions for the Volterra lattice hierarchy

Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers

We derive an explicit formula for the connected $(n,m)$-point functions associated to an arbitrary diagonal tau-function $\tau_f(\boldsymbol{t}^+,\boldsymbol{t}^-)$ of the 2d Toda lattice hierarchy using fermionic computations and the boson-fermion correspondence. Then for fixed $\boldsymbol{t}^-$, we compute the KP-affine coordinates of $\tau_f(\boldsymbol{t}^+,\boldsymbol{t}^-)$. As applications, we present a unified approach to compute various types of connected double Hurwitz numbers, including the ordinary double Hurwitz numbers, the double Hurwitz numbers with completed $r$-cycles, and the mixed double Hurwitz numbers. We also apply this method to the computation of the stationary Gromov-Witten invariants of $\mathbb P^1$ relative to two points.

Multiple Skew-Orthogonal Polynomials and 2-Component Pfaff Lattice Hierarchy

Multiple Skew-Orthogonal Polynomials and 2-Component Pfaff Lattice Hierarchy

A generalized integrable lattice hierarchy related to the Ablowitz–Ladik lattice: Conservation law, Darboux transformation and exact solution

Under investigation in this paper is a more general discrete [Formula: see text] matrix spectral problem. Starting from this spectral problem, the positive and negative integrable lattice hierarchies are constructed based on the Tu scheme, then by considering linear combination of the positive and negative lattice hierarchies, we give a more general integrable lattice hierarchy, which can reduce to the well-known Ablowitz–Ladik lattice and the discrete modified Korteweg–de Vries (mKdV) equation. In particular, we obtain some local and nonlocal integrable lattice equations, including reverse-space discrete mKdV equation, reverse-space complex discrete mKdV equation, higher-order discrete mKdV equation, higher-order complex discrete mKdV equation, higher-order reverse-space discrete mKdV equation and higher-order reverse-space complex discrete mKdV equation. In additional, infinitely many conservation laws and Darboux transformation (DT) for the first non-trivial system in the hierarchy are established with the help of its Lax pair. The exact solutions of the system are generated by applying the obtained DT. The results in this paper might be helpful for understanding some physical phenomena.

Solitons and normal random matrices

We discuss a general relation between the solitons and statistical mechanics and show that the partition function of the normal random matrix model can be obtained from the multi-soliton solutions of the two-dimensional Toda lattice hierarchy in a special limit.

Solving the (2+1)-Dimensional Derivative Toda Equation

The main objectives of this work are to deduce the Lax pair for the Modified Toda lattice hierarchy and to find whether there exists a transformation between (2+1)-dimensional derivative Toda lattice equation and Modified Toda equation. By a new spectral problem, we obtain the Modified Toda lattice hierarchy, furthermore we give the Hirota’s form solution for the Modified Toda lattice equation, thus the Hirota’s solution for the (2+1)-dimensional derivative Toda equation can also be attained correspondingly.

Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy

The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer N, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande’s dressing operators for the equivariant Gromov-Witten theory of . This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to 0. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.

Spectral curves and W-representations of matrix models

We explain how the spectral curve can be extracted from the mathcal{W} -representation of a matrix model. It emerges from the part of the mathcal{W} -operator, which is linear in time-variables. A possibility of extracting the spectral curve in this way is important because there are models where matrix integrals are not yet available, and still they possess all their important features. We apply this reasoning to the family of WLZZ models and discuss additional peculiarities which appear for the non-negative value of the family parameter n, when the model depends on additional couplings (dual times). In this case, the relation between topological and 1/N expansions is broken. On the other hand, all the WLZZ partition functions are τ-functions of the Toda lattice hierarchy, and these models also celebrate the superintegrability properties.

Grothendieck’s dessins d’enfants in a web of dualities. III

We give a new proof of the equivalence between the dessin partition function and the partition function of the Laguerre unitary ensemble (LUE), originally found by Ambjørn and Chekhov. We also introduce a correction factor for the dessin/LUE partition function, and show that the corrected dessin/LUE partition function is a Dubrovin–Zhang type tau-function of the Toda lattice hierarchy. As an application, we use the approach of Dubrovin and Zhang for the computation of the dessin correlators. In physicists’ terminology, we establish dualities among dessin counting, generalized Penner model, and -topological sigma model.

Darboux transformaiton and exact solutions of a four-component Volterra lattice system

In this letter, by means of the discrete zero curvature representation, four-component Volterra lattice (FCVL) hierarchy and its first flow, i.e., FCVL system are presented from a 4 × 4 matrix spectral problem with four potential functions. Moreover, a Darboux transformation for the FCVL system is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problems, by which some exact solutions of the FCVL system are obtained in different cases.

Integrable lattice hierarchies behind Cauchy two-matrix model and Bures ensemble

This paper focuses on different reductions of the two-dimensional (2d)-Toda hierarchy. Symmetric and skew-symmetric moment matrices are first considered, resulting in differential relations between symmetric/skew-symmetric tau functions and 2d-Toda’s tau functions, respectively. Furthermore, motivated by the Cauchy two-matrix model and Bures ensemble from random matrix theory, we study the rank-one shift condition in the symmetric case and rank-two shift condition in the skew-symmetric case, from which the C-Toda and B-Toda hierarchies are determined, together with their special Lax matrices and integrable structures.

Finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy

Resorting to the zero-curvature equation and the Lenard recursion equations, the generalized Merola–Ragnisco–Tu lattice hierarchy associated with a 3 × 3 discrete matrix spectral problem is derived. With the aid of the characteristic polynomial of the Lax matrix for the generalized Merola–Ragnisco–Tu lattice hierarchy, a trigonal curve is defined, on which we construct the Baker–Akhiezer function, two meromorphic functions, three kinds of Abelian differentials, and Riemann theta function. By analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions, especially their asymptotic expansions near three infinite points and three zero points, we obtain their essential singularities and divisors. Finally, we obtain the finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy in terms of the Riemann theta function.

Liouville integrability and explicit solutions for a Frobenius‐coupled integrable lattice hierarchy

We define a Frobenius‐coupled integrable lattice hierarchy including its Lax pair. Then the bi‐Hamiltonian structures of this hierarchy are constructed, which show that this hierarchy possesses Liouville integrability. We also obtained N‐fold Darboux transformation of the Frobenius‐coupled integrable lattice hierarchy. Finally, the explicit solutions after onefold Darboux transformation of this hierarchy are investigated when we choose appropriate seed solutions.

The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy

We extend the matrix-resolvent method of computing logarithmic derivatives of tau-functions to the nonlinear Schrödinger (NLS) hierarchy. Based on this method we give a detailed proof of a theorem of Carlet, Dubrovin and Zhang regarding the relationship between the Toda lattice hierarchy and the NLS hierarchy. As an application, we give an improvement of an algorithm of computing correlators in hermitian matrix models.

Lax Operator Algebras and Applications to τ-Symmetries for Multilayer Integrable Couplings

The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As illustrative examples, isospectral and non-isospectral integrable couplings and the corresponding commutator relations of their Lax operators are computed explicitly in the cases of the Volterra lattice hierarchy and the AKNS hierarchy, along with their τ-symmetry algebras.

Three-sheeted Riemann surface and solutions of the Itoh–Narita–Bogoyavlensky lattice hierarchy

The Itoh–Narita–Bogoyavlensky lattice hierarchy associated with a discrete [Formula: see text] matrix spectral problem is derived by using Lenard recursion equations. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a three-sheeted Riemann surface [Formula: see text] of arithmetic genus [Formula: see text] and construct the corresponding Baker–Akhiezer function and meromorphic function on it. On the basis of the theory of Riemann surface, the continuous flow and discrete flow related to the lattice hierarchy are straightened with the help of the Abel map. Quasi-periodic solutions of the lattice hierarchy in terms of the Riemann theta function are constructed by using the asymptotic properties and the algebro-geometric characters of the meromorphic function and Riemann surface.

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

Dynamical self-organization and efficient representation of space by grid cells.

To plan trajectories and navigate, animals must maintain a mental representation of the environment and their own position within it. This "cognitive map" is thought to be supported in part by the entorhinal cortex, where grid cells are active when an animal occupies the vertices of a scaling hierarchy of periodic lattices of locations in an enclosure. Here, we review computational developments which suggest that the grid cell network is: (a) efficient, providing required spatial resolution with a minimum number of neurons, (b) self-organizing, dynamically coordinating the structure and scale of the responses, and (c) adaptive, re-organizing in response to changes in landmarks and the structure of the boundaries of spaces. We consider these ideas in light of recent discoveries of similar structures in the mental representation of abstract spaces of shapes and smells, and in other brain areas, and highlight promising directions for future research.