  In this article, we define the notion of $n$-cotorsion pairs in triangulated
categories, which is a generalization of the classical cotorsion pairs. We
prove that any mutation of an $n$-cotorsion pair is again an $n$-cotorsion
pair. When $n=1$, this result generalizes the work of Zhou and Zhu for
classical cotorsion pairs. As applications, we give a geometric
characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A$
and give a geometric realization of mutation of $n$-cotorsion pairs via
rotation of certain configurations of $n$-diagonals.
