Periodic and rational solutions of modified Korteweg-de Vries equation

Abstract: We present closed form periodic solutions of the integrable modified Korteweg-de Vriesequation (mKdV). By using a Darboux transformation, we derive first-and second-orderdoubly-periodic lattice-like solutions. We explicitly derive first-and second-orderrational solutions as limiting cases of periodic solutions. We have also found thedegenerate solution which corresponds to the equal eigenvalue case. Among the second-ordersolutions, we single out the doubly-localized high peak solution on a constant backgroundwith an infinitely extended trough. This solution plays the role of a rogue wave of themKdV equation. Graphical abstract: [Figure not available: see fulltext.]