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We study the long-time dynamics of complex-valued modified Korteweg-de Vries(mKdV) solitons, which are recognized because they blow-up in finite time. Weestablish stability properties at the H^1 level of regularity, uniformly awayfrom each blow-up point. These new properties are used to prove that mKdVbreathers are H^1 stable, improving our previous result, where we only provedH^2 stability. The main new ingredient of the proof is the use of a B\"acklundtransformation which links the behavior of breathers, complex-valued solitonsand small real-valued solutions of the mKdV equation. We also prove thatnegative energy breathers are asymptotically stable. Since we do not use anymethod relying on the Inverse Scattering Transformation, our proof works evenunder rough perturbations, provided a corresponding local well-posedness theoryis available.

Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers