Lax pairs for 3D spatial problems based on the Dirac operator

Abstract

In this paper, we show that Lax pairs can be constructed using the Dirac operator in the context of Clifford analysis. Since Lax pairs are closely linked to spectral decompositions they are not easily obtainable in the context of Dirac operators due to the non-commutativity of the underlying algebraic structure. The main idea is to substitute the classic Lax approach by the so-called AKNS method. We demonstrate that it is possible to obtain Lax pairs for both linear and non-linear PDE’s in this way.