Scalar parabolic PDEs and braids

• Published in: Transactions of the American Mathematical Society 2009-05, Vol.361 (5), p.2755-2788
• Main Authors: Ghrist, R. W., Vandervorst, R. C.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Braiding; Critical points; Exact sciences and technology; Functional analysis; General logic; Global analysis, analysis on manifolds; Logic and foundations; Mathematical analysis; Mathematical logic, foundations, set theory; Mathematics; Partial differential equations; Periodic orbits; Polynomials; Recurrence relations; Research article; Scalars; Sciences and techniques of general use; Skeleton; Topology; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-08-04823-X

The comparison principle for scalar second order parabolic PDEs on functions u(t,x)u(t,x) admits a topological interpretation: pairs of solutions, u1(t,⋅)u^1(t,\cdot ) and u2(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 {uα(t,⋅)}α=1n\{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 uα(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.