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Reversible variational partial differential equations such as the Swift–Hohenberg equation can admit localized stationary roll structures whose solution branches are bounded in parameter space but unbounded in function space, with the width of the roll plateaus increasing without bound along the branch: this scenario is commonly referred to as snaking. In this work, the structure of the bifurcation diagrams of localized rolls is investigated for variational but non-reversible systems, and conditions are derived that guarantee snaking or result in diagrams that either consist entirely of isolas. [ABSTRACT FROM PUBLISHER]

Partial differential equations with time delays serve as models for systems where both spatial structure and memory effects are important. The asymptotic stability of travelling waves in such systems is still determined entirely by the spectrum of the linearization about the wave. We compare the spectra of localized waves on the unbounded real line with spectra computed on large intervals with appropriate boundary conditions applied at their end points. We show that the spectrum on large intervals approximates the spectrum on the real line when periodic boundary conditions are used. If separated boundary conditions are applied, it is the so-called absolute spectrum together with the extended point spectrum that is approximated; their union typically differs from the spectrum on the real line. [ABSTRACT FROM AUTHOR]