A twisted Marchenko lemma and applications

We develop a twisted Marchenko lemma with applications into non-commutative two-component integrable systems. Starting from a twisted differential equation, we introduce the corresponding two-component Wronski matrix $W_{n,m}(\mathit{\phi },\mathit{\psi })$. Under the Marchenko lemma conditions, the logarithmic derivative $(\mathrm{\partial }{W}_{n,m})W_{n,m}^{-1}$ satisfies nonlinear equations, whose two-component projections yield a non-commutative two-component KP equation, while a single-component reduction recovers the non-commutative Burgers equation, KP equation and DS equation. A direct approach based on two-component quasi-Wronskians is developed, from which a higher-order non-commutative two-component KP equation is generated with matrix-valued solutions. Reductions of the seed functions of the non-commutative two-component KP hierarchy yield the one-component counterpart, while a stepwise reduction of those for the two-component KP equation gives the non-commutative DS and NLS equations.