Growth and heterogeneities in pattern formation and invasion fronts
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Abstract
We consider three different situations. First, we explore transverse pattern formation in the Cahn-Hilliard equation in the wake of a quench. We pose this problem on an infinite channel, and through a quenching heterogeneity prove that there exist two one-parameter families bifurcating from a solution to the Cahn-Hilliard equation arising from an $O(2)$-Hopf bifurcation as the quench speed varies. We then investigate various properties of these bifurcating branches, including preliminary investigations into stability and some numerical investigations into what happens as the transverse wavenumber varies, as well as what happens as the state from which the branches bifurcate is varied. We next consider fronts in the presence of an increasing, slowly-varying parameter. We determine the leading-order front position, speed, and steepness of the front in the presence of this parameter. We then provide some numerical explorations into a delayed invasion problem. We also determine that, for patterns forming fronts in the complex Ginzburg-Landau equation, an inviscid Burger's equation provides a good approximation of the wavenumber at the leading edge of the front. We then move on to consider quasipattern formation. We give an overview of quasipatterns, as well as collecting some of the rigorous work which has been done regarding the formation of quasipatterns. Finally, we explore a beginning to extending these rigorous results to additional equations, as well as conducting some numerical investigations.
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2025