We consider radial solutions to the cubic Schrödinger equation on the Heisenberg groupThis equation is a model for totally non-dispersive evolution equations. We show existence of ground state traveling waves with speed β ∈ (−1, 1). When the speed β is sufficiently close to 1, we prove their uniqueness up to symmetries and their smoothness along the parameter β. The main ingredient is the emergence of a limiting system as β tends to the limit 1, for which we establish linear stability of the ground state traveling wave.

We consider the third order Benjamin-Ono equation on the torusWe prove that for any t ∈ R, the flow map continuously extends to H s r,0 (T) if s ≥ 0, but does not admit a continuous extension to H −s r,0 (T) if 0 < s < 1 2 . Moreover, we show that the extension is not weakly sequentially continuous in L 2 r,0 (T). We then classify the traveling wave solutions for the third order Benjamin-Ono equation in L 2 r,0 (T) and study their orbital stability.

The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov, where the pathological set contains a superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schrödinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity, which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales.