Reduction of the dressing chain of the Schrödinger operator

We reduce the problem of constructing real finite-gap solutions of the focusing modified Korteweg–de Vries equation to the dressing chain of the Schrödinger operator. We show that the Schrödinger operator spectral curve corresponding to such a solution is real. We give some restrictions on the initial data for the chain that lead to such solutions. We also consider a soliton reduction. We obtain compact representations for the multisoliton and breather solutions of the modified Korteweg–de Vries equationentations can be useful in developing the perturbation theory for various applied problems.