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A wide variety of stationary or moving spatially localized structures ispresent in evolution problems on unbounded domains, governed byhigher-than-second-order reversible spatial interactions. This work provides ageneric unfolding in one spatial dimension of a certain codimension-threesingularity that explains the organization of bifurcation diagrams of suchlocalized states in a variety of contexts, ranging from nonlinear optics tofluid mechanics, mathematical biology and beyond. The singularity occurs when acusp bifurcation associated with the onset of bistability between homogeneoussteady states encounters a pattern-forming, or Turing, bifurcation. The lattercorresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamicsproblem. Such codimension-three points are sometimes called Lifshitz points inthe physics literature. In the simplest case where the spatial system conservesa first integral, the system is described by a canonical fourth order scalarsystem. The problem contains three small parameters, two that unfold the cuspbifurcation and one that unfolds the Turing bifurcation. Several cases arerevealed, depending on open conditions on the signs of the lowest-ordernonlinear terms. Taking the case in which the Turing bifurcation issubcritical, various parameter regimes are considered and the bifurcationdiagrams of localized structures are elucidated. A rich bifurcation structureis revealed, which involves transitions between regions of localized periodicpatterns generated by homoclinic snaking, and mesa-like patterns with uniformcores. The theory is shown to unify previous numerical results obtained inmodels arising in nonlinear optics, fluid mechanics, and excitable media moregenerally.

Organization of spatially localized structures near a codimension-three cusp-Turing bifurcation