  This paper deals with the study of the behaviour of the wave functions of a
two-component Bose-Einstein condensate near the interface, in the case of
strong segregation. This yields a system of two coupled ODE's for which we want
to have estimates on the asymptotic behaviour, as the strength of the coupling
tends to infinity. As in phase separation models, the leading order profile is
a hyperbolic tangent. We construct an approximate solution and use the
properties of the associated linearized operator to perturb it into a genuine
solution for which we have an asymptotic expansion. We prove that the
constructed heteroclinic solutions are linearly nondegenerate, in the natural
sense, and that there is a spectral gap, independent of the large interaction
parameter, between the zero eigenvalue (due to translations) at the bottom of
the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness
result which implies that, in fact, the constructed heteroclinic is the unique
minimizer (modulo translations) of the associated energy, for which we provide
an expansion.
