Or equivalently

Introduction
In this paper, we study the breather solutions of the following negative order modified Korteweg-de Vries (nmKdV) equation
(1.1)
(1.2)
or equivalently
(1.3)
We call it “negative order” as it is in the negative flow of the mKdV hierarchy[1]. Some recent studies about the nmKdV equation can be found in [1-3]. In recent years, the negative order equations attract much attention from researchers[4-8]. Many physically meaningful systems, such as the Camassa-Holm equation[9,10], the Degasperis-Procesi equation[11], and the short pulse equation[12-14], are associated to negative order equations through reciprocal transformations.
There are several well-known equations that are closely related to the nmKdV equations (1.1)-(1.2) or (1.3), for example, the mKdV equation, the Gardner equation and the sine-Gordon equation. Those equation all have breather solutions. As far as we know, breather solution is a kind of wave solution that is of soliton structures and related to lump solutions[15-19].