| Summary: | Understanding the characteristics of interface motion and front propagation is an important feature in many scientific areas such as invasive species, avalanche, combustion, solidification and many other industrial processes. This thesis is concerned with introducing, investigating, solving and discussing some models for the front instabilities that suit the shapes of fronts observed in applications better than the Kuramoto-Sivashinsky equation. Our attention has been focused on dealing with three systems that have the growth rate proportional to |k| for small wavenumber k, where the front dynamics takes different shapes such as lobe-and-cleft patterns. These systems are the nonlocal Kuramoto-Sivashinsky (nonlocal KS), Michelson-Sivashinsky (MS) and modified Michelson-Sivashinsky (MMS) equations.
In this work, we examine all three systems numerically in one and two dimensions, varying the domain size. For a small domain in one dimension, the dynamics of the front for all three systems show coarsening from initial disturbances until it reaches a stable steady-state solution with one cusp. The dynamics of the front for the MS and MMS equations in a large domain coarsens from initial disturbances until it reaches a state with a few cusps. Then this state seems to be unstable and new small cusps appear in the troughs and move toward the large cusps. The dynamics of the front for the nonlocal-KS equation in a large domain shows similar behaviour, but with a small nonlocal term is fully unstable with no coherent structure. In addition, travelling wave and heteroclinic cycle solutions between the large and small domains have been found for the nonlocal-KS equation. The dynamics of the front in two dimensions has similar behaviour to the one-dimensional case for all three systems. The most novel finding of this work is a family of analytical solutions of the MS equation in one and two dimensions.
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