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Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity. (English) Zbl 1483.35119

Summary: 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.

MSC:

35K58 Semilinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B44 Blow-up in context of PDEs
37C29 Homoclinic and heteroclinic orbits for dynamical systems

Software:

INTLAB; GitHub
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Full Text: DOI arXiv

References:

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