From one to infinity: symmetries of integrable systems

TL;DR

This paper demonstrates that identifying a single nonlocal key-symmetry in integrable systems like Korteweg-de Vries and Boussinesq equations suffices to establish their integrability and derive their infinite symmetries and Lax pairs.

Contribution

It introduces a novel approach where only one nonlocal key-symmetry is needed to confirm integrability and generate the entire symmetry structure of the system.

Findings

A single nonlocal key-symmetry guarantees integrability.

From the key-symmetry, recursion operators and infinite symmetries are constructed.

The method simplifies the process of verifying integrability.

Abstract

Integrable systems constitute an essential part of modern physics. Traditionally, to approve a model is integrable one has to find its infinitely many symmetries or conserved quantities. In this letter, taking the well known Korteweg-de Vries and Boussinesq equations as examples, we show that it is enough to find only one nonlocal key-symmetry to guarantee the integrability. Starting from the nonlocal key-symmetry, recursion operator(s) and then infinitely many symmetries and Lax pairs can be successfully found.