Periodic and rational solutions of modified Korteweg-de Vries equation

We present closed form periodic solutions of the integrable modified Korteweg-de Vries equation (mKdV). By using a Darboux transformation, we derive first-and second-order doubly-periodic lattice-like solutions. We explicitly derive first-and second-order rational solutions as limiting cases of periodic solutions. We have also found the degenerate solution which corresponds to the equal eigenvalue case. Among the second-order solutions, we single out the doubly- localized high peak solution on a constant background with an infinitely extended trough. This solution plays the role of a rogue wave of the mKdV equation.