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In this paper, we present a novel center manifold reduction theorem forquasilinear elliptic equations posed on infinite cylinders. This is donewithout a phase space in the sense that we avoid explicitly reformulating thePDE as an evolution problem. Under suitable hypotheses, the resulting centermanifold is finite dimensional and captures all sufficiently small boundedsolutions. Compared with classical methods, the reduced ODE on the manifold ismore directly related to the original physical problem and also easier tocompute. The analysis is conducted directly in H\"older spaces, which is oftendesirable for elliptic equations.  We then use this machinery to construct small bounded solutions to a varietyof systems. These include heteroclinic and homoclinic solutions of theanti-plane shear problem from nonlinear elasticity; exact slow moving invasionfronts in a two-dimensional Fisher--KPP equation; and hydrodynamic bores withvorticity in a channel. The last example is particularly interesting in that wefind solutions with critical layers and distinctive "half cat's eye" streamlinepatterns.

Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics