We prove that the solution map associated with the $1D$ half-wave cubic equation in the periodic setting cannot be uniformly continuous on bounded sets of the periodic Sobolev spaces $H^s$ with $s\in ( \frac 14 , \frac 12 )$.
Vladimir Georgiev. Nikolay Tzvetkov. Nicola Visciglia. "On the regularity of the flow map associated with the 1D cubic periodic Half-Wave equation." Differential Integral Equations 29 (1/2) 183 - 200, January/February 2016. https://doi.org/10.57262/die/1448323259