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In this paper we study high order expansions of chart maps for local finitedimensional unstable manifolds of hyperbolic equilibrium solutions of scalarparabolic partial differential equations. Our approach is based on studying aninfinitesimal invariance equation for the chart map that recovers the dynamicson the manifold in terms of a simple conjugacy. We develop formal seriessolutions for the invariance equation and efficient numerical methods forcomputing the series coefficients to any desired finite order. We show, undermild non-resonance conditions, that the formal series expansion converges in asmall enough neighborhood of the equilibrium. An a-posteriori computer assistedargument proves convergence in larger neighborhoods. We implement the methodfor a spatially inhomogeneous Fisher's equation and numerically compute andvalidate high order expansions of some local unstable manifolds for morse indexone and two. We also provide a computer assisted existence proof of asaddle-to-sink heteroclinic connecting orbit.

Fourier-Taylor Parameterization of Unstable Manifolds for Parabolic  Partial Differential Equations: Formalization, Implementation, and Rigorous  Validation