Scalar parabolic PDE's and braids

The comparison principle for scalar second order parabolic PDEs on functions $u(t,x)$ admits a topological interpretation: pairs of solutions, $u^1(t,\cdot)$ and $u^2(t,\cdot)$, evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^\alpha(t,\cdot)\}_{\alpha=1}^n$. By lifting the graphs to Legendrian braids, we give a global version of the comparison principle: the curves $u^\alpha(t,\cdot)$ 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.