A number of stationary localized solutions to the well-known pattern-forming gradient system mentioned in the title have been found. Their search is based on the theory of homoclinic orbits to a saddle-focus equilibrium and some results of linear symmetric differential operators with decaying coefficients along with computer simulations.
Belyakov, L. A., Glebsky, L. Y., & Lerman, L. M. (1997). Abundance of stable stationary localized solutions to the generalized 1D Swift-Hohenberg equation. Computers and Mathematics with Applications, 34(2–4), 253–266. https://doi.org/10.1016/s0898-1221(97)00127-2