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Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings

Vincent Caudrelier, Peter H. van der Kamp, Cheng Zhang

TL;DR

The paper tackles the problem of constructing discrete integrable mappings from quad-equation systems with boundaries. It introduces open boundary reductions on a strip, leveraging dual boundary equations and a discrete boundary zero curvature condition to obtain a double-row monodromy matrix whose trace yields invariants. Explicit examples for H1 and Q1$(oldsymbol{ abla}=0)$ demonstrate low-dimensional maps with concrete invariants, including a non-QRT planar map arising from a 3D autonomous/non-autonomous reduction. This framework provides an alternative to periodic reductions, offering new insights into integrability with boundaries and potential generalizations to other lattices and boundary geometries with implications for discrete Liouville integrability and beyond-ABS equations.

Abstract

In the context of integrable partial difference equations on quad-graphs, we introduce the notion of open boundary reductions as a new means to construct discrete integrable mappings and their invariants. This represents an alternative to the well-known periodic reductions. The construction deals with well-posed initial value problems for quad equations on quad-graphs restricted to a strip. It relies on the so-called double-row monodromy matrix and gives rise to integrable mappings. To obtain the double-row monodromy matrix, we use the notion of boundary matrix and discrete boundary zero curvature condition, themselves related to the boundary consistency condition, which complements the well-known $3$D consistency condition for integrable quad equations and gives an integrability criterion for boundary equations. This relation is made precise in this paper. Our focus is on quad equations from the Adler-Bobenko-Suris (ABS) classification and their discrete integrable boundary equations. Taking as our prime example a regular $\mathbb{Z}^2$-lattice with two parallel boundaries, we provide an explicit construction of the maps obtained by open boundary reductions, the boundary matrices, as well as the invariants extracted from the double-row monodromy matrix. Examples of integrable maps are considered for the H1 and Q1($δ=0$) equations and an interesting example of a non-QRT map of the plane is presented. Examples of well-posed quad-graph systems on a strip beyond the $\mathbb{Z}^2$-lattice are also given.

Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings

TL;DR

The paper tackles the problem of constructing discrete integrable mappings from quad-equation systems with boundaries. It introduces open boundary reductions on a strip, leveraging dual boundary equations and a discrete boundary zero curvature condition to obtain a double-row monodromy matrix whose trace yields invariants. Explicit examples for H1 and Q1 demonstrate low-dimensional maps with concrete invariants, including a non-QRT planar map arising from a 3D autonomous/non-autonomous reduction. This framework provides an alternative to periodic reductions, offering new insights into integrability with boundaries and potential generalizations to other lattices and boundary geometries with implications for discrete Liouville integrability and beyond-ABS equations.

Abstract

In the context of integrable partial difference equations on quad-graphs, we introduce the notion of open boundary reductions as a new means to construct discrete integrable mappings and their invariants. This represents an alternative to the well-known periodic reductions. The construction deals with well-posed initial value problems for quad equations on quad-graphs restricted to a strip. It relies on the so-called double-row monodromy matrix and gives rise to integrable mappings. To obtain the double-row monodromy matrix, we use the notion of boundary matrix and discrete boundary zero curvature condition, themselves related to the boundary consistency condition, which complements the well-known D consistency condition for integrable quad equations and gives an integrability criterion for boundary equations. This relation is made precise in this paper. Our focus is on quad equations from the Adler-Bobenko-Suris (ABS) classification and their discrete integrable boundary equations. Taking as our prime example a regular -lattice with two parallel boundaries, we provide an explicit construction of the maps obtained by open boundary reductions, the boundary matrices, as well as the invariants extracted from the double-row monodromy matrix. Examples of integrable maps are considered for the H1 and Q1() equations and an interesting example of a non-QRT map of the plane is presented. Examples of well-posed quad-graph systems on a strip beyond the -lattice are also given.

Paper Structure

This paper contains 17 sections, 5 theorems, 103 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Boundary equations, folding and discrete boundary zero curvature conditions
  3. $3$D-consistent quad equations and discrete zero curvature conditions
  4. Boundary consistency and integrable boundary equations
  5. Folding and dual boundary equations
  6. Dual boundary consistency and the boundary zero curvature condition
  7. Examples of integrable boundary equations and boundary matrices
  8. H1, case of additive $\sigma$
  9. Q1($\delta=0$), case of additive $\sigma$
  10. Q1($\delta=0$), case of multiplicative $\sigma$
  11. Integrable mappings from open boundary reductions
  12. Open boundary reductions on the ${\mathbb Z}^2$-lattice
  13. Generating function for the invariants
  14. Examples of open boundary reductions for the H1 and Q1($\delta=0$) equations
  15. H1 additive case with two different boundary equations
  16. ...and 2 more sections

Key Result

Lemma 2

Let $Q=Q(x,y,c,z;\alpha,\beta)$ be a multi-linear function with ${\cal D}_4$ symmetry, and let the boundary equation (eq:generalform) possess properties (i)-(iii). Suppose that $p=p(y,x,c;\alpha,\beta)$ is a divisor of $q_1Q(x,y,c, -\frac{q_2}{q_1};\alpha,\beta)$ and that $p$ also possesses propert Then, we have: $$

Figures (15)

  • Figure 1: A quad-graph with a boundary consisting of triangular faces
  • Figure 2: Elementary quadrilateral supporting the bulk equation.
  • Figure 3: $3$D consistency: a quad equation can be imposed on the six faces of a cube. The black dots indicate the initial values.
  • Figure 4: Elementary triangle supporting a boundary equation. The thick line represents the boundary connecting the boundary fields.
  • Figure 5: Boundary consistency around half of a rhombic dodecahedron (left) and its planar projection (right), where $Q=0$ is imposed on $4$ quadrilaterals and $q=0$ is imposed on $4$ triangles.
  • ...and 10 more figures

Theorems & Definitions (8)

  • Definition 1
  • Lemma 2: Duality
  • Definition 3
  • Definition 4
  • Proposition 5
  • Theorem 6
  • Corollary 7
  • Corollary 8