Svetlana Roudenko: An angle between past and future: from soliton stability in Zakharov–Kuznetsov equation to fractio…

Abstract:

In this talk we start with the original 3d Zakharov–Kuznetsov (ZK) equation and examine stability of solitary waves. In particular, we prove the asymptotic stability, which is based on Liouville-type theorem that uses a la virial operator and heavily relies on its spectral properties, which we obtain from an Angle lemma. We then briefly examine the 2d version of ZK in a critical setting (cubic nonlinearity) and prove the existence of blow-up, again by understanding spectral properties with the angle lemma. When examining the solutions to the ZK equation, we notice that the radiation is emitted in an angle. This angle depends on the dispersion, thus, allowing us to study the generalization of ZK to fractional KdV-type equations including higher-dimensional Benjamin-Ono models, where we also investigate stability and instability of solitary waves.