SU(N) q-Toda equations from mass deformed ABJM theory

It is known that the partition functions of the U(N)k × U(N + M)−k ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painlevé III3 equation. In this paper we have suggested that a similar bilinear relation holds for the ABJM theory with $$ \mathcal{N} $$ N = 6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition function for various N, k, M and the mass parameter. For particular choices of the mass parameters labeled by integers ν, a as m1 = m2 = −πi(ν − 2a)/ν, the bilinear relation corresponds to the q-deformation of the affine SU(ν) Toda equation in τ-form.

Explicit Solutions of Integrable Variable-coeffcient Cylindrical Toda Equations

Integrable cylindrical Toda lattice equations are proposed by utilizing a generalized version of the dressing method. A compatibility condition is given which insures that these equations are integrable. Further, soliton solutions for new type equations are shown in explicit forms, including one soliton solution and two soliton solutions, respectively.

A q-generalization of the Toda equations for the q-Laguerre/Hermite orthogonal polynomials

Based on the motivation of generalizing the correspondence between the Lax equation for the Toda lattice and the deformation theory of the orthogonal polynomials, we derive a [Formula: see text]-deformed version of the Toda equations for both [Formula: see text]-Laguerre/Hermite ensembles, and check the compatibility with the quadratic relation.

Integrable Properties of a Variant of the Discrete Hungry Toda Equations and Their Relationship to Eigenpairs of Band Matrices

The Toda equation and its variants are studied in the filed of integrable systems. One particularly generalized time discretisation of the Toda equation is known as the discrete hungry Toda (dhToda) equation, which has two main variants referred to as the dhTodaI equation and dhTodaII equation. The dhToda equations have both been shown to be applicable to the computation of eigenvalues of totally nonnegative (TN) matrices, which are matrices without negative minors. The dhTodaI equation has been investigated with respect to the properties of integrable systems, but the dhTodaII equation has not. Explicit solutions using determinants and matrix representations called Lax pairs are often considered as symbolic properties of discrete integrable systems. In this paper, we clarify the determinant solution and Lax pair of the dhTodaII equation by focusing on an infinite sequence. We show that the resulting determinant solution firmly covers the general solution to the dhTodaII equation, and provide an asymptotic analysis of the general solution as discrete-time variable goes to infinity.