| Abstract: |
| In this talk I am going to present various results regarding the
derivation of amplitude equations for nonlocal PDEs. In particular, we
shall consider nonlocal reaction terms as well as nonlocal fractional
diffusion in the context of the Swift-Hohenberg equation.
Interestingly,
the resulting Ginzburg-Landau-type amplitude equations become fully
local with the non-local influence only entering to leading-order in
the
coefficients. This shows an interesting decoupling effect near
instability. I shall explain, how this effect arises in the proofs as
well. The work is based upon the following papers: (1) Validity of
amplitude equations for nonlocal nonlinearities, C. Kuehn and S.
Throm,
Journal of Mathematical Physics, Vol. 59, 071510, 2018. (2) The
amplitude equation for the space-fractional Swift-Hohenberg equation,
C. Kuehn and S. Throm, Physica D, Vol. 472, 134531, 2025. |
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