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
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.
Key words: Nonlinear Dynamics
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2016