A new integrable equation valued on a Cayley-Dickson algebra

TL;DR

This paper introduces a novel integrable equation valued on Cayley-Dickson algebras, generalizing the KdV equation, with explicit conserved quantities, solutions, and symmetries, including for octonions.

Contribution

It presents the first integrable equation on Cayley-Dickson algebras, with a Bäcklund transformation, Lax pair, and soliton solutions, extending known results for quaternions.

Findings

Equation reduces to KdV when algebra is complex

Infinite conserved quantities for each Cayley-Dickson algebra

Explicit soliton solutions and symmetries for the new equation

Abstract

We introduce a new integrable equation valued on a Cayley-Dickson (C-D) algebra. In the particular case in which the algebra reduces to the complex one the new interacting term in the equation cancells and the equation becomes the known Korteweg-de Vries equation. For each C-D algebra the equation has an infinite sequence of local conserved quantities. We obtain a B\"{a}cklund transformation in the sense of Walhquist-Estabrook for the equation for any Cayley-Dickson algebra, and relate it to a generalized Gardner equation. From it, the infinite sequence of conserved quantities follows directly. We give the explicit expression for the first few of them. From the B\"{a}cklund transformation we get the Lax pair and the one-soliton and two-soliton solutions generalizing the known solutions for the quaternion valued KdV equation. From the Gardner equation we obtain the generalized modified KdV equation which also has an infinite sequence of conserved quantities. The new integrable equation is preserved under a subgroup of the automorphisms of the C-D algebra. In the particular case of the algebra of octonions, the equation is invariant under $SU(3)$.