Centrally Extended Conformal Galilei Algebras and Invariant Nonlinear PDEs

We construct, for any given $\ell = \frac{1}{2} + {\mathbb N}_0,$ the second-order \textit{nonlinear} partial differential equations (PDEs) which are invariant under the transformations generated by the centrally extended conformal Galilei algebras. The generators are obtained by a coset construction and the PDEs are constructed by standard Lie symmetry technique. It is observed that the invariant PDEs have significant difference for $\ell > \frac{3}{2}.$