  Consider the scattering of a time-harmonic acoustic plane wave by a bounded
elastic obstacle which is immersed in a homogeneous acoustic medium. This paper
concerns an inverse acoustic-elastic interaction problem, which is to determine
the location and shape of the elastic obstacle by using either the phased or
phaseless far-field data. By introducing the Helmholtz decomposition, the model
problem is reduced to a coupled boundary value problem of the Helmholtz
equations. The jump relations are studied for the second derivatives of the
single-layer potential in order to establish the corresponding boundary
integral equations. The well-posedness is discussed for the solution of the
coupled boundary integral equations. An efficient and high order
Nystr\"{o}m-type discretization method is proposed for the integral system. A
numerical method of nonlinear integral equations is developed for the inverse
problem. For the case of phaseless data, we show that the modulus of the
far-field pattern is invariant under a translation of the obstacle. To break
the translation invariance, an elastic reference ball technique is introduced.
We prove that the inverse problem with phaseless far-field pattern has a unique
solution under certain conditions. In addition, a numerical method of the
reference ball technique based nonlinear integral equations is also proposed
for the phaseless inverse problem. Numerical experiments are provided to
demonstrate the effectiveness and robustness of the proposed methods.
