Jordan algebras and integrable systems (Q1337912)

We construct Jordan analogs of the scalar mKdV and sG equations. It is shown that every system \[ w^i_t= w^i_{xxx}- 6a^i_{jk} w^j w^k_x,\quad i= 1,\dots, N\tag{1} \] admits the differential substitution \[ w^i= u^i_x+ a^i_{jk} u^j u^k,\quad i= 1,\dots, N,\tag{2} \] which is a natural generalization of the Miura map. By substituting (2) into (1) we obtain the Jordan mKdV system \[ u^i_t= u^i_{xxx}- 6a^i_{nm} a^n_{jk} u^j u^k u^m_x,\quad i= 1,\dots, N. \] We also construct Jordan analogs of the scalar pmKdV equation.