TY - JOUR

T1 - Linear Integrable Systems on Quad-Graphs

AU - Bobenko, Alexander I.

AU - Suris, Yuri B.

N1 - Publisher Copyright:
© The Author(s) 2021. Published by Oxford University Press. All rights reserved.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - In the first part of the paper, we classify linear integrable (multidimensionally consistent) quad-equations on bipartite isoradial quad-graphs in C, enjoying natural symmetries and the property that the restriction of their solutions to the black vertices satisfies a Laplace type equation. The classification reduces to solving a functional equation. Under certain restriction, we give a complete solution of the functional equation, which is expressed in terms of elliptic functions. We find two real analytic reductions, corresponding to the cases when the underlying complex torus is of a rectangular type or of a rhombic type. The solution corresponding to the rectangular type was previously found by Boutillier, de Tilière, and Raschel. Using the multidimensional consistency, we construct the discrete exponential function, which serves as a basis of solutions of the quad-equation. In the second part of the paper, we focus on the integrability of discrete linear variational problems. We consider discrete pluriharmonic functions, corresponding to a discrete two-form with a quadratic dependence on the fields at black vertices only. In an important particular case, we show that the problem reduces to a two-field generalization of the classical star-triangle map. We prove the integrability of this novel 3D system by showing its multidimensional consistency. The Laplacians from the first part come as a special solution of the two-field star-triangle map.

AB - In the first part of the paper, we classify linear integrable (multidimensionally consistent) quad-equations on bipartite isoradial quad-graphs in C, enjoying natural symmetries and the property that the restriction of their solutions to the black vertices satisfies a Laplace type equation. The classification reduces to solving a functional equation. Under certain restriction, we give a complete solution of the functional equation, which is expressed in terms of elliptic functions. We find two real analytic reductions, corresponding to the cases when the underlying complex torus is of a rectangular type or of a rhombic type. The solution corresponding to the rectangular type was previously found by Boutillier, de Tilière, and Raschel. Using the multidimensional consistency, we construct the discrete exponential function, which serves as a basis of solutions of the quad-equation. In the second part of the paper, we focus on the integrability of discrete linear variational problems. We consider discrete pluriharmonic functions, corresponding to a discrete two-form with a quadratic dependence on the fields at black vertices only. In an important particular case, we show that the problem reduces to a two-field generalization of the classical star-triangle map. We prove the integrability of this novel 3D system by showing its multidimensional consistency. The Laplacians from the first part come as a special solution of the two-field star-triangle map.

UR - http://www.scopus.com/inward/record.url?scp=85158901960&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnab151

DO - 10.1093/imrn/rnab151

M3 - Article

AN - SCOPUS:85158901960

SN - 1073-7928

VL - 2022

SP - 14639

EP - 14674

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 19

ER -