The multi-component second modified Korteweg–de Vries equation and its two distinct integrable coupling types

A new isospectral problem is designed and the multi-component second mKdV equation is worked out from it. It follows that two distinct types of integrable couplings of the multi-component second mKdV equation are obtained by constructing two types of new loop algebras. As its reduction, two distinct types of integrable couplings of the multi-component KdV equation, the multi-component mKdV equation and the multi-component KdV–mKdV equation are presented.