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 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 transform the solution to the simplest 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.