  In this paper, a partial proof of a conjecture raised by Galaktionov and
Svirshchevskii concerning existence and global uniqueness of an asymptotically
stable periodic orbit in a fourth-order piecewise linear ordinary differential
equation is presented. The fourth-order equation comes from the study of
traveling wave patterns in a signed Kuramoto-Sivashinsky equation with
absorption. The proof is twofold. First, the problem of solving for the
periodic orbit is transformed into a zero finding problem in four dimensional
space, which is solved with a computer-assisted proof based on Newton's method
and the contraction mapping theorem. Second, the rigorous bounds about the
periodic orbit in phase space are combined with the theory of discontinuous
dynamical systems to prove that the orbit is asymptotically stable.
