Topic
Method of matched asymptotic expansions
About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.
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TL;DR: In this paper, the authors derived a nonlinear-diffusion porous-Fisher's equation for population dynamics using explicit traveling wave solutions, initially-separated, expanding populations are studied as they first coalesce.
50 citations
TL;DR: In this paper, the authors analyzed the spreading of a drop on a perfectly smooth solid surface using the method of matched asymptotic expansions, including the effect of intermolecular forces near the contact line both on surface diffusion of adsorbed molecules and on flow within the liquid phase itself near the surface line.
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Journal Article•
TL;DR: In this paper, the solitary-wave solutions of Benjamin's model were investigated for a class of equations that include Benjamin's equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension.
Abstract: Benjamin recently put forward a model equation for the evolution of waves on the interface of a two-layer system of fluids in which surface tension effects are not negligible. In this case, the fluid motion η on the interface of these two fluids can be approximately described by an equation ηt + ηx + ηηx − αLηx ± βηxxx = 0, where η depends on saptial variable x and time variable t, and L = H∂x is the composition of the Hilbert transform and the spatial derivative in the direction of primary propagation, or, equivalently, L is a Fourier multiplier operator with symbol |ξ|. It is our purpose here to investigate the solitary-wave solutions of Benjamin’s model. For a class of equations that include Benjamin’s equation, which feature conflicting contributions to dispersion from dynamic effects on the interface and surface tension, we establish existence of travelling-wave solutions. This is complished by using P.L. Lions concentrated-compactness principle. Using the recently developed theory of Li and Bona, we are also able to determine rigorously the spatial asymptotics of these solutions. Department of Mathematics, The University of Texas at Austin, Austin, TX 78712. Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, Austin, TX 78712.
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TL;DR: In this paper, it was shown that if the leading-order spin and quadrupole moment vanish, then through second order in its mass it moves on a geodesic of a smooth, locally causal vacuum metric defined in its local neighbourhood.
Abstract: When a small, uncharged, compact object is immersed in an external background spacetime, at zeroth order in its mass it moves as a test particle in the background. At linear order, its own gravitational field alters the geometry around it, and it moves instead as a test particle in a certain effective metric satisfying the linearized vacuum Einstein equation. In the letter [Phys. Rev. Lett. 109, 051101 (2012)], using a method of matched asymptotic expansions, I showed that the same statement holds true at second order: if the object's leading-order spin and quadrupole moment vanish, then through second order in its mass it moves on a geodesic of a certain smooth, locally causal vacuum metric defined in its local neighbourhood. Here I present the complete details of the derivation of that result. In addition, I extend the result, which had previously been derived in gauges smoothly related to Lorenz, to a class of highly regular gauges that should be optimal for numerical self-force computations.
50 citations
TL;DR: Degond and Raviart as mentioned in this paper provided a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter e is set equal to zero) and the boundary layer problem.
Abstract: 187 Degond, P. and P.A. Raviart, An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the Child-Langmuir law, Asymptotic Analysis 4 (1991) 187-214. We perform the asymptotic analysis of the one-dimensional Vlasov-Poisson system when singular boundary data are prescribed. Such a singular perturbation problem arises in the modelling of vacuum diodes under very large applied bias, and gives rise to the well-known "Child-Langmuir law". In this paper, we provide a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter e is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.
50 citations