# Multicomplex solitons

## TL;DR

This paper explores extensions of nonlinear wave equations to multicomplex algebras, analyzing multi-soliton solutions and symmetries like $ ext{PT}$-symmetry, revealing new qualitative behaviors and conserved quantity properties.

## Contribution

It introduces multicomplex, quaternionic, coquaternionic, and octonionic versions of integrable wave equations, demonstrating how symmetries influence solutions and conserved quantities.

## Key findings

- Multi-soliton solutions exhibit novel qualitative behaviors.
- Symmetries enforce certain conserved quantities to be real.
- Noncommutative equations relate to $ ext{PT}$-symmetry in multicomplex units.

## Abstract

We discuss integrable extensions of real nonlinear wave equations with multi-soliton solutions, to their bicomplex, quaternionic, coquaternionic and octonionic versions. In particular, we investigate these variants for the local and nonlocal Korteweg-de Vries equation and elaborate on how multi-soliton solutions with various types of novel qualitative behaviour can be constructed. Corresponding to the different multicomplex units in these extensions, real, hyperbolic or imaginary, the wave equations and their solutions exhibit multiple versions of antilinear or $\mathcal{PT}$-symmetries. Utilizing these symmetries forces certain components of the conserved quantities to vanish, so that one may enforce them to be real. We find that symmetrizing the noncommutative equations is equivalent to imposing a $\mathcal{PT}$-symmetry for a newly defined imaginary unit from combinations of imaginary and hyperbolic units in the canonical representation.