Toda 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.

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.

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.

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.

General rogue wave solution to the discrete nonlinear Schrödinger equation

In the present paper, we are concerned with the rogue wave solution to an integrable discrete nonlinear Schrödinger (NLS) equation. First, through the KP–Toda reduction method, the general breather solution of the discrete NLS equation is deduced from two bilinear equations in discrete two-dimensional Toda-lattice hierarchy. Then, by taking an ingenious and successive limit to the breather solution, we derive the higher order rogue wave solution to the discrete NLS equation.

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.

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.

Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model

We consider solutions of the 2D Toda lattice hierarchy which are elliptic functions of the zeroth time t_0=x. It is known that their poles as functions of t_1 move as particles of the elliptic Ruijsenaars-Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians which govern the dynamics of poles with respect to the m-th hierarchical times t_m and \bar t_m of the 2D Toda lattice hierarchy are obtained from expansion of the spectral curve for the Lax matrix of the Ruijsenaars-Schneider model at the marked points.

ON THE INTEGRATION OF A HIRERARCHY FOR THE PEREODIC TODA LATTICE WITH A FREE TERM

In this article, we have explored the Toda lattice hierarchy in the class of periodic functions with a free term. We have given an effective method of constructing of the periodic Toda lattice hierarchy with a free term. We have discussed the complete integrability of the constructed systems that is based on the inverse spectral problem of an associated discrete Hill`s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy.

A generalized integrable lattice hierarchy associated with the Toda and modified Toda lattice equations: Hamiltonian representation, soliton solutions

In this paper, we give an integrable four-field lattice hierarchy associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. The Hamiltonian representation of the four-field lattice hierarchy are constructed by using the trace identity. We consider reductions of our hierarchy, and find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, and some new integrable three-field lattice hierarchies. In addition, the modified Toda lattice equation is studied, the figures of one-, two-soliton and periodic solutions with properly parameters are shown graphically to illustrate the propagation of solitary waves, and we find that the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The elastic and inelastic interactions among the two-soliton for the modified Toda lattice equation are also discussed. Furthermore, we generate the relationships between the structures of exact solutions and parameters, and present soliton solutions with free parameters. The results in this paper might be helpful for interpreting certain physical phenomena.

Toda lattice hierarchy and soliton equations on square lattice

Lattice soliton equations of KdV-family and Boussinesq-family are investigated from the viewpoint of the Toda lattice hierarchy. As a consequence, special solutions such as soliton solutions and algebro-geometric solutions are constructed in a unified manner.

Matrix Resolvent and the Discrete KdV Hierarchy

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered.

On tau-functions for the Toda lattice hierarchy

We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the generating series of $k$-point correlation functions of the solution. Applications to computing GUE correlators and Gromov--Witten invariants of the Riemann sphere are under consideration.

ABJM matrix model and 2D Toda lattice hierarchy

It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions.

Decomposing the Toda hierarchy by an implicit symmetry constraint

An implicit symmetry constraint of the famous Toda lattice hierarchy is presented. Using this symmetry constraint, every lattice equation in the Toda hierarchy is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

Integration of the Toda lattice hierarchy with self-consistent source via inverse scattering method

Integration of the Toda lattice hierarchy with self-consistent source via inverse scattering method

Non-Abelian integrable hierarchies: matrix biorthogonal polynomials and perturbations

In this paper, Geronimus–Uvarov perturbations for matrix orthogonal polynomials on the real line are studied and then applied to the analysis of non-Abelian integrable hierarchies. The orthogonality is understood in full generality, i.e. in terms of a nondegenerate continuous sesquilinear form, determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. We derive Christoffel-type formulas that give the perturbed matrix biorthogonal polynomials and their norms in terms of the original ones. The keystone for this finding is the Gauss–Borel factorization of the Gram matrix. Geronimus–Uvarov transformations are considered in the context of the 2D non-Abelian Toda lattice and noncommutative KP hierarchies. The interplay between transformations and integrable flows is discussed. Miwa shifts, τ-ratio matrix functions and Sato formulas are given. Bilinear identities, involving Geronimus–Uvarov transformations, first for the Baker functions, then secondly for the biorthogonal polynomials and its second kind functions, and finally for the τ-ratio matrix functions, are found.

Generating series for GUE correlators

We extend to the Toda lattice hierarchy the approach of Bertola et al. (Phys D Nonlinear Phenom 327:30–57, 2016; IMRN, 2016) to computation of logarithmic derivatives of tau-functions in terms of the so-called matrix resolvents of the corresponding difference Lax operator. As a particular application we obtain explicit generating series for connected GUE correlators. On this basis an efficient recursive procedure for computing the correlators in full genera is developed.