Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

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