Bright and dark breathers on an elliptic wave in the defocusing mKdV equation

TL;DR

This paper constructs new bright and dark breather solutions on an elliptic wave background for the defocusing mKdV equation, solving a longstanding open problem by developing a novel eigenfunction representation.

Contribution

Introduces a new eigenfunction representation of the Lax operator, enabling the explicit construction of breather solutions on elliptic backgrounds for the defocusing mKdV.

Findings

Constructed two families of breathers with bright and dark profiles.

Resolved the open problem of breather construction on elliptic wave backgrounds.

Provided explicit formulas for breather solutions in the defocusing mKdV context.

Abstract

Breathers on an elliptic wave background consist of nonlinear superpositions of a soliton and a periodic wave, both traveling with different wave speeds and interacting periodically in the space-time. For the defocusing modified Korteweg-de Vries (mKdV) equation, the construction of general breathers has been an open problem since the elliptic wave is related to the elliptic degeneration of the hyperelliptic solutions of genus two. We have found the new representation of eigenfunctions of the Lax operator associated with the elliptic wave, which enables us to solve this open problem and to construct two families of breathers with bright (elevation) and dark (depression) profiles.