On consistency of determinants on cubic lattices

On Initial Data in the Problem of Consistency on Cubic Lattices for $3 \times 3$ Determinants

The paper is devoted to complete proofs of theorems on consistency on cubic lattices for $3 \times 3$ determinants. The discrete nonlinear equations on $\mathbb{Z}^2$ defined by the condition that the determinants of all $3 \times 3$ matrices of values of the scalar field at the points of the lattice $\mathbb{Z}^2$ that form elementary $3 \times 3$ squares vanish are considered; some explicit concrete conditions of general position on initial data are formulated; and for arbitrary initial data satisfying these concrete conditions of general position, theorems on consistency on cubic lattices (…

Consistency on cubic lattices for determinants of arbitrary orders

We consider a special class of two-dimensional discrete equations defined by relations on elementary NxN squares, N>2, of the square lattice Z^2, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary NxN squares, N>2, in the cubic lattice Z^3. For an arbitrary N we prove such consistency on cubic lattices for two-dimensional discrete equations defined by the condition that the determinants of values of the field at the points of the square lattice Z^2 that are contained in elementary NxN squares vanish.

Asymptotics of a cubic sine kernel determinant

We study the one parameter family of Fredholm determinants $\det(I-γK_{\textnormal{csin}}),γ\in\mathbb{R}$ of an integrable Fredholm operator $K_{\textnormal{csin}}$ acting on the interval $(-s,s)$ whose kernel is a cubic generalization of the sine kernel which appears in random matrix theory. This Fredholm determinant appears in the description of the Fermi distribution of semiclassical non-equilibrium Fermi states in condensed matter physics as well as in random matrix theory. Using the Riemann-Hilbert method, we calculate the large $s$-asymptotics of $\det(I-γK_{\textnormal{csin}})$ for all…

Discrete integrable equations on face-centered cubics: consistency and Lax pairs of corner equations

A new set of discrete integrable equations, called face-centered quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang-Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centered-cube (CAFCC), which requires consistency of an overdetermined system of fourteen five-point equations on the face-centered cubic unit cell. In this paper a new formulation of CAFCC is introduced where so-called type-C equations are centered at faces of the face-centered cubic unit cel…

具有三次非线性的修正Camassa-Holm方程的可积半离散化

Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

本文基于离散KP方程和离散逆变换，通过约化过程和引入因变量，导出了具有三次非线性的修正Camassa-Holm方程的可积半离散化，并证明其在连续极限下收敛于连续方程。

The inverse spectral problem for the discrete cubic string

Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-ϕ'''=zmϕ$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis--Procesi equation. We solve the spectral and inverse spectral problem for…