  We study the linear profile decomposition for the Airy type equation, where
the associated Strichartz inequality corresponds to the Fourier extension
inequality on the odd curve $\xi^{\ell}$. We also investigate an inhomogeneous
case, modeled by the odd curve $\xi^3+\xi^5$ case. We note that, as observed by
Frank and Sabin [Math. Ann., 2018], there is a two-profile phenomenon in the
profile decomposition associated with odd curves.
