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

TL;DR

This paper demonstrates the existence of heteroclinic orbits and unbounded solutions in complex-valued evolutionary PDEs with quadratic nonlinearities using computer-assisted proofs, advancing understanding of their dynamics.

Contribution

It introduces a rigorous numerical framework to prove heteroclinic connections and unbounded solutions in complex PDEs, which was previously unestablished.

Findings

Existence of heteroclinic orbits from nontrivial equilibria to zero.

Unbounded solutions along unstable manifolds at equilibrium.

Validation of trapping regions around the zero equilibrium.

Abstract

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.