Abstract
The functionalized Cahn--Hilliard (FCH) free energy models phase separation in amphiphilic systems, which possess a component with both hydrophilic and hydrophobic tendencies. We establish the existence of a family of bilayer structures, described by homoclinic solutions of a fourth-order dynamical system arising from the FCH energy in the nongeneric case of an equal-depth double well, $W\geq0$. The free parameter accommodates bilayers of variable width, which correspond to the compression of the bilayer under external pressure. Modulo a nondegeneracy condition, which we conjecture is associated to an orbit-flip bifurcation within the fourth-order dynamical system, we show that its homoclinic orbits are well approximated by those of a corresponding second-order dynamical system obtained by perturbing the potential well $W$ with the primitive of Modica and Mortola's $\sqrt{W}$.
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