  We study the existence of patterns (nontrivial, stationary solutions) for
one-dimensional Swift-Hohenberg Equation in a directional quenching scenario,
that is, on $x\leq 0$ the energy potential associated to the equation is
bistable, whereas on $x\geq 0$ it is monostable. This heterogeneity in the
medium induces a symmetry break that makes the existence of heteroclinic orbits
of 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 in
one-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 a
new approach in finding connecting orbits, using what we call `far/near
decompositions', relying both on information about the spatial behavior of the
solutions and on Fourier analysis. Our method is functional analytic and PDE
based, relying minimally on dynamical system techniques and making no use of
comparison principles whatsoever.
