Scalar parabolic PDEs and braids

Ghrist, R.; Vandervorst, R.

doi:10.1090/S0002-9947-08-04823-X

Scalar parabolic PDEs and braids
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by R. W. Ghrist and R. C. Vandervorst PDF

Trans. Amer. Math. Soc. 361 (2009), 2755-2788 Request permission

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.

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