  At T = 0 and a sufficiently large field, the nearest-neighbor
antiferromagnetic Ising chain undergoes a first-order spin-flop transition into
the ferromagnetic phase. We consider its smearing under the random-bond
disorder such that all independent random bonds are antiferromagnetic (AF). It
is shown that it can be described exactly for arbitrary distribution of AF
bonds P(J). Moreover, the site magnetizations of finite chains can be found
analytically in this model. We consider continuous P(J) which is zero above
some -J_1 and behaves near it as (-J_1 - J)^\lambda, \lambda > -1. In this case
ferromagnetic phase emerges continuously in a field H > H_c = 2J_1. At 0 >
\lambda > -1 it has usual second-order anomalies near H_c with critical indices
obeying the scaling relation and depending on \lambda . At \lambda > 0 the
higher-order transitions appear (third, fourth etc.) marked by the divergence
of corresponding nonlinear susceptibilities. In the chains with even number of
spins the intermediate "bow-tie" phase with linearly modulated AF order exists
between AF and ferromagnetic ones at J_1 < H < H_c. Its origin can be traced to
the infinite correlation length of the degenerate AF phase from which it
emerges. This implies the existence of similar inhomogeneous phases with size-
and form-dependent order in a number of other systems with infinite correlation
length. The possibility to observe the signs of "bow-tie" phase in low-T
neutron diffraction experiments is discussed.
