Rigorous theory for slanted snaking in the conserved Swift–Hohenberg equation

Develop a rigorous existence and bifurcation theory for slanted snaking of stationary localized solutions to the conserved Swift–Hohenberg (phase‑field crystal) equation U_t = -Δ[ -(q^2+Δ)^2 U - μ U + ν U^2 - U^3 ] on finite spatial domains, including characterization of the snaking region, its dependence on the conserved mass and domain size, and the associated amplitude‑equation reductions.

Background

In mass‑conserving variants of the Swift–Hohenberg equation posed on bounded domains, stationary localized states organize along ‘slanted’ snaking curves rather than the standard snaking observed in non‑conserved systems.

While weakly nonlinear expansions and numerical continuation capture the slanted geometry, a general rigorous framework establishing existence, parameter dependence, and bifurcation structure for slanted snaking in the conserved Swift–Hohenberg equation is missing.