We present a rigorous numerical method for proving the existence of a localised radially symmetric solution for a Ginzburg-Landau type equation. This has a direct application to the problem of finding spots in the Swift-Hohenberg equation. The method is more generally applicable to finding radially symmetric solutions of stationary PDEs on the entire space. One can rewrite such a problem in the form of a singular ODE. We transform this ODE to a finite domain and use a Green's function approach to formulate an appropriate integral equation. We then construct a mapping whose fixed points coincide with solutions to the ODE, and show via computeraided analytic estimates that the mapping is contracting on a small neighbourhood of a numerically determined approximate solution.
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