A search for integrable evolution equations with Lax pairs over the octonions

TL;DR

This paper discovers four new integrable evolution equations with Lax pairs over octonions, expanding the understanding of integrability in non-associative algebraic structures.

Contribution

It introduces a method to find integrable equations with Lax pairs over octonions using a scaling ansatz and formulates conditions for Lax pairs in this setting.

Findings

Four new integrable equations with Lax pairs over octonions

A systematic method for constructing such equations

Conditions for differential operators to form Lax pairs over octonions

Abstract

Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV scaling weights. A condition for linear differential operators to be a Lax pair over octonions is formulated and solved for the unknown coefficients in the polynomials.