  We consider the problem of determining the initial state of integrable
quantum field theory quenches in terms of the post-quench eigenstates. The
corresponding overlaps are a fundamental input to most exact methods to treat
integrable quantum quenches. We construct and examine an infinite integral
equation hierarchy based on the form factor bootstrap, proposed earlier as a
set of conditions deter- mining the overlaps. Using quenches of the mass and
interaction in Sinh-Gordon theory as a concrete example, we present theoretical
arguments that the state has the squeezed coherent form expected for integrable
quenches, and supporting an Ansatz for the solution of the hierarchy. Moreover
we also develop an iterative method to solve numerically the lowest equation of
the hierarchy. The iterative solution along with extensive numerical checks
performed using the next equation of the hierarchy provide a strong numerical
evidence that the proposed Ansatz gives a very good approximation for the
solution.
