University of Birmingham > Talks@bham > Applied Mathematics Seminar Series > Exponential asymptotics and homoclinic snaking in continuous and discrete systems

Exponential asymptotics and homoclinic snaking in continuous and discrete systems

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Homoclinic snaking of localized patterns has been observed in a variety of experimental and theoretical contexts. The phenomenon, in which a multiplicity of localized states exists within an exponentially small parameter range, is due to a slowly varying amplitude ’locking’ to the underlying, fast-scale pattern. Through a careful asymptotic analysis of the one-dimensional Swift-Hohenberg equation, we show how the conventional method of multiple scales near bifurcation must be extended to incorporate exponentially small effects if a complete asymptotic description of snaking behaviour is to be achieved. We then apply similar techniques to study one-dimensional snaking on a square lattice, in which the slow amplitude locks onto the spatial grid, and show that the snaking region is non-zero only when the solution is oriented at an angle which has a rational or infinite tangent.

This talk is part of the Applied Mathematics Seminar Series series.

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