In this paper we study various properties of the double ramification
hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15]
using intersection theory of the double ramification cycle in the moduli space
of stable curves. In particular, we prove a recursion formula that recovers the
full hierarchy starting from just one of the Hamiltonians, the one associated
to the first descendant of the unit of a cohomological field theory. Moreover,
we introduce analogues of the topological recursion relations and the divisor
equation both for the hamiltonian densities and for the string solution of the
double ramification hierarchy. This machinery is very efficient and we apply it
to various computations for the trivial and Hodge cohomological field theories,
and for the r-spin Witten's classes. Moreover we prove the Miura equivalence
between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for
the Gromov-Witten theory of the complex projective line (extended Toda
hierarchy).Comment: Revised version, to be published in Communications in Mathematical
Physics, 27 page
