  In this paper, we consider the dynamics of solutions to complex-valued
evolutionary partial differential equations (PDEs) and show existence of
heteroclinic orbits from nontrivial equilibria to zero via computer-assisted
proofs. We also show that the existence of unbounded solutions along unstable
manifolds at the equilibrium follows from the existence of heteroclinic orbits.
Our computer-assisted proof consists of three separate techniques of rigorous
numerics: an enclosure of a local unstable manifold at the equilibria, a
rigorous integration of PDEs, and a constructive validation of a trapping
region around the zero equilibrium.
