From nonassociativity to solutions of the KP hierarchy

AbstractA recently observed relation between ‘weakly nonassociative’ algebras $$\mathbb{A}$$ (for which the associator ( $$\mathbb{A},\mathbb{A}^2 ,\mathbb{A}$$ ) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus $$\mathbb{A}$$ ′ of { $$\mathbb{A}$$ ) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with matrix algebra $$\mathbb{A}$$ ′, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.