Multicomplex solitons

Abstract

We discuss integrable extensions of real nonlinear wave equations with multisoliton 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 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 PT -symmetry for a newly defined imaginary unit from combinations of imaginary and hyperbolic units in the canonical representation.

Nonlinear Classical and Quantum Integrable Systems with PT-symmetries

“…Similar to the previous chapter, we will investigate here further extensions of the real KdV equation not in the complex [33,30], but the multicomplex regime [35]. These are higher order complex extensions, in particular they will be of bicomplex, quaternionic, coquaternionic and octonionic types.…”

“…After carrying out work in the complex regime, a good question to ask at this point is, what are the results and properties of higher order complex extensions, for example with bicomplex, quaternionic, coquaternionic and octonionic types? In Chapter 4 [35],…”

Lie sphere geometry in nuclear scattering processes

“…The bicomplex number can be further extended to multicomplex numbers, see Price [31]. An application of multicomplex numbers in physics has been proposed recently in [32]. The bicomplex number has one real and three imaginary parts…”