We study the long-time dynamics of complex-valued modified
Korteweg–de Vries (mKdV) solitons, which are distinguished because
they blow up in finite time. We establish stability properties at the
level of regularity, uniformly away from each blow-up point.
These new properties are used to prove that mKdV breathers are
-stable,
improving our previous result [Comm. Math. Phys. 324:1 (2013) 233–262], where we only
proved
-stability.
The main new ingredient of the proof is the use of a Bäcklund transformation
which relates the behavior of breathers, complex-valued solitons and small
real-valued solutions of the mKdV equation. We also prove that negative
energy breathers are asymptotically stable. Since we do not use any method
relying on the inverse scattering transform, our proof works even under
perturbations, provided a corresponding local well-posedness theory is available.
