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We study the existence of patterns (nontrivial, stationary solutions) forone-dimensional Swift-Hohenberg Equation in a directional quenching scenario,that is, on $x\leq 0$ the energy potential associated to the equation isbistable, whereas on $x\geq 0$ it is monostable. This heterogeneity in themedium induces a symmetry break that makes the existence of heteroclinic orbitsof the type point-to-periodic not only plausible but, as we prove here, true.In this search, we use an interesting result of "Fefferman, Charles, J.Lee-Thorp, and Michael Weinstein. Topologically protected states inone-dimensional systems. Vol. 247. No. 1173. American Mathematical Society,2017" in order to understand the multiscale structure of the problem, namely,how fast/slow scales interact with each other. In passing, we advocate for anew approach in finding connecting orbits, using what we call `far/neardecompositions', relying both on information about the spatial behavior of thesolutions and on Fourier analysis. Our method is functional analytic and PDEbased, relying minimally on dynamical system techniques and making no use ofcomparison principles whatsoever.

The Swift-Hohenberg Equation under directional-quenching: finding heteroclinic connections using spatial and spectral decompositions