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

Abstract

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 $H^1$ level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are $H^1$-stable, improving our previous result [Comm. Math. Phys. 324:1 (2013) 233–262], where we only proved $H^2$-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 $L^2\left(ℝ\right)$ perturbations, provided a corresponding local well-posedness theory is available.