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.The authors acknowledge the support of the Australian Research Council (Discovery Project number DP140100265). N.A. and A.A. acknowledge support from the Volkswagen Stiftung and A.C. acknowledges Endeavour Postgraduate Award support and support from a Ben Williams student grant