Scalar parabolic PDEs and braids

Abstract

The comparison principle for scalar second order parabolic PDEs on functions u ( t , x ) u(t,x) admits a topological interpretation: pairs of solutions, u 1 ( t , ⋅ ) u^1(t,\cdot ) and u 2 ( t , ⋅ ) 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 α ( t , ⋅ ) } α = 1 n \{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 α ( t , ⋅ ) 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.