Eur. Phys. J. D (2016) 70: 104
https://doi.org/10.1140/epjd/e2016-70033-9

Regular Article

Periodic and rational solutions of modified Korteweg-de Vries equation

Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2600, Australia

^a e-mail: huqemdad@gmail.com

Received: 18 January 2016
Received in final form: 9 March 2016
Published online: 12 May 2016

Abstract

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.