Multi-component generalizations of mKdV equation and nonassociative algebraic structures

Abstract

Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg–de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie–Jordan algebras [Speciality of Lie–Jordan algebras, J. Algebra 237 (2001) 621–636].

Drinfel'd-Sokolov construction and exact solutions of vector modified KdV hierarchy

“…for the complex field v = u 1 + iu 2 . Equation (1) appeared in [3,4] in the study of the evolution of a curve in a (N +1)dimensional Riemannian manifold, while several generalisations of the mKdV equation have been introduced by various authors such as, for example, Iwao and Hirota [30], Fordy and Athorne [11] in association with symmetric spaces, Sokolov and Wolf [36] using the symmetry approach, Svinolupov and Sokolov [8,9] in relation to Jordan algebras, and non-associative algebras in general [35]. Moreover, the study of soliton solutions and their interactions for several multi-component generalisations of the scalar mKdV equation have been studied using the inverse scattering transform, or the Hirota, dressing, or other methods, see for example [6,30,31,32,33,34].…”