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Scalar parabolic PDE's and braids | | R. Ghrist
; R. C. Vandervorst
; | | Date: |
18 Mar 2004 | | Subject: | Dynamical Systems; Geometric Topology | math.DS math.GT | | Abstract: | 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. | | Source: | arXiv, math.DS/0403308 | | Services: | Forum | Review | PDF | Favorites | |
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