A. Mucalica and D.E. Pelinovsky
Dark breathers on a snoidal wave background in the defocusing MKDV equation
Abstract:
We present a new exact solution to the defocusing modified Korteweg-de Vries equation
to describe the interaction of a dark soliton and a traveling periodic wave.
The solution (which we refer to as to the dark breather) is obtained by using the Darboux
transformation with the eigenfunctions of the Lax system expressed in terms of the Jacobi
theta functions. Properties of elliptic functions including the quarter-period translations
in the complex plane are applied to simplify the solution to a closed form. We explore
the characteristic properties of these dark breathers and show that they propagate faster
than the periodic wave (in the same direction) and attain maximal localization at a
specific parameter value which is explicitly computed.
Keywords:
modified Korteweg-de Vries equation, snoidal traveling waves, dark breathers, Darboux transformation, Jacobi theta functions.