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In the talk below I discuss the construction of heteroclinic connections in Swift-Hohenberg Equations under directional quenching (DQ). I briefly mention the techniques used - mostly from harmonic and functional analysis - which we apply to decompose the spatial and spectral structure of the pattern to understand its multiscale nature. The simulation below depicts what we commonly see in the directional quenching context: on the wake of a quenching front (the"trigger" that you see moving to the left) we observe a clear manifestation of a self-organized phenomenon.
I should emphasize that the talk is somewhat different from the paper: later on, I realized that our result was stronger than we initially thought. This helped a lot to clarify the analysis and the notation (with that said, please check the paper :).
The patterns discussed in this talk concern the case of a fixed quenching front; nonetheless, they are related to the ones seen on the wake of the front in the above simulation:
The Swift-Hohenberg Equation under directional-quenching: finding heteroclinic connections using spatial and spectral decompositions (USA 2020; note: simulation videos not included)
Directional quenching, when applied to other pattern forming equations like Cahn-Hilliard or Allen-Cahn, also give interesting phenomena:
In joint work with Arnd Scheel, many results for DQ in Allen-Cahn and Cahn-Hilliard were obtained (many interesting open problems were left behind though!). The two talks below discuss some of these results:
Multi-dimensional patterns from directional quenching (USA - 2018)
Phase-separations patterns from directional quenching (Japan - 2017)
The poster of the conference "Shock waves and beyond", held at the Institut Henri Poincaré (Paris) in 2015 features a figure from the work that B. Barker, K.Zumbrun, and I published years later. Kevin Zumbrun gave a talk about this work in Bordeaux-France (link below).
Corrugation instability and bifurcation of slow viscous MHD shocks in a duct (by Kevin Zumbrun)
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