arXiv subjects

Benjamin F. Akers

Publications and source records attributed to Benjamin F. Akers.

3 recordsLinked to original sources

arXiv2025

Coherent Structures in Flame Fronts

We study traveling waves in a coordinate-free model of flame fronts. The flame front is the interface between the burnt and unburnt phases of a gas undergoing combustion. The front therefore moves in a preferred direction, as the unburnt gas is consumed. In the horizontally periodic, vertically unbounded setting, we prove the existence of waves of permanent form which are traveling in the vertical direction. We also compute these waves. The analysis and computation use the framework of traveling waves in the arclength parameterization as previously developed by two of the authors and Wright.

math.AP↗

arXiv2018

Periodic traveling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic traveling waves on infinite depth, and computed such traveling waves. The formulation of the traveling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of traveling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.

math.AP↗

arXiv2017

Periodic traveling interfacial hydroelastic waves with or without mass

We study the motion of an interface between two irrotational, incompressible fluids, with elastic bending forces present; this is the hydroelastic wave problem. We prove a global bifurcation theorem for the existence of families of spatially periodic traveling waves on infinite depth. Our traveling wave formulation uses a parameterized curve, in which the waves are able to have multi-valued height. This formulation and the presence of the elastic bending terms allows for the application of an abstract global bifurcation theorem of "identity plus compact" type. We furthermore perform numerical computations of these families of traveling waves, finding that, depending on the choice of parameters, the curves of traveling waves can either be unbounded, reconnect to trivial solutions, or end with a wave which has a self-intersection. Our analytical and computational methods are able to treat in a unified way the cases of positive or zero mass density along the sheet, the cases of single-valued or multi-valued height, and the cases of single-fluid or interfacial waves.

math.AP↗