A Search for Integrable Evolution Equations with Lax Pairs over the Octonions

Abstract

Lax pairs play a vital role in the integrability theory of evolution equations because they are used for the inverse scattering transformation to generate multi-soliton solutions. As an extension of integrable real evolution equations and their known Lax pair, this report focuses on the search for octonion evolution equations u_t = F(u, u_x, u_xx, u_xxx) of KdV type and mKdV type that have a Lax pair, where u(t, x) is an octonion variable. A Lax pair is defined as L_t ψ = M(Lψ) − L(Mψ) with linear differential operators L and M whose coefficients depend on u and x-derivatives of u, where ψ(t, x) is an auxiliary octonion function. These operators act on ψ by producing a linear polynomial in ψ and x-derivatives of ψ such that each term is a product involving u, u_x, . . . , and (x-derivatives of) ψ in a given order. It is assumed that the evolution equation u_t = F(u, u_x, u_xx, u_xxx) as well as both Lψ and Mψ are homogeneous under a scaling of t, x, u which is either the scaling associated to the KdV equation or the mKdV equation. This leads to an overdetermined system of algebraic equations for the (real-valued) coefficients of u and x−derivatives of u in F, Lψ and Mψ. The formulation of the overdetermined system involves two important differences compared to the case of a real variable u. Firstly, since octonions are non-associative and non-commutative, F, Lψ and Mψ contain many more terms, with different orderings of products. In particular, ψ (and its x-derivatives) are allowed to appear on the left, in the middle, or on the right. Secondly, products of octonions obey certain algebraic identities, whereby terms that are equivalent modulo these identities must be eliminated. To solve the overdetermined system, Maple is used to do the splittings, and depending on the complexity of the system, ’rifsimp’ in Maple or a package called ’Crack’ in Reduce are used to solve it. As a main result, a single KdV octonion equation, three mKdV octonion equations, and also a single potential-KdV octonion equation, each of which has more than one Lax pair, are obtained.