SCALAR PARABOLIC PDES AND BRAIDS

The comparison principle for scalar second order parabolic PDEs on functions u(t,x) admits a topological interpretation: pairs of solutions, u(1)(t,.) and u(2)(t,.), evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions {u(alpha)(t,.)}(alpha=1)(n). By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves u(alpha)(t, .) evolve so as to (weakly) decrease the algebraic length of the braid. We define a Morse-type theory on Legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic PDEs. This is done via discretization to a finite-dimensional system and a suitable Conley index for discrete braids. The result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic PDEs. We give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time-periodic solutions.