The linear profile decomposition for the Airy equation and the existence   of maximizers for the Airy Strichartz inequality
  In this paper, we establish the linear profile decomposition for the Airy equation with complex or real initial data in $L^2$, respectively. As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy-Strichartz inequality. 
Shuanglin Shao