From nothing to something: discrete integrable systems

TL;DR

This paper derives various discrete integrable systems from fundamental principles, suggesting their inherent integrability due to simple algebraic identities and geometric invariants, bridging concepts from physics, mathematics, and computer science.

Contribution

It introduces a novel approach to generate discrete integrable models from 'nothing' using basic algebraic and geometric principles, unifying diverse systems under a common framework.

Findings

Derived well-known integrable systems like KdV, KP, Toda from fundamental principles.

Proposed that these models are inherently integrable due to their algebraic and geometric origins.

Established conjecture linking discrete models to simple algebraic identities and geometric invariants.

Abstract

Chinese ancient sage Laozi said that everything comes from `nothing'. Einstein believes the principle of nature is simple. Quantum physics proves that the world is discrete. And computer science takes continuous systems as discrete ones. This report is devoted to deriving a number of discrete models, including well-known integrable systems such as the KdV, KP, Toda, BKP, CKP, and special Viallet equations, from `nothing' via simple principles. It is conjectured that the discrete models generated from nothing may be integrable because they are identities of simple algebra, model-independent nonlinear superpositions of a trivial integrable system (Riccati equation), index homogeneous decompositions of the simplest geometric theorem (the angle bisector theorem), as well as the M\"obious transformation invariants.