Abstract
We study dynamics of interfaces in solutions of the equation \( \varepsilon \Box u + \frac{1}{\varepsilon }f_\varepsilon (u)=0\), for \(f_\varepsilon \) of the form \(f_\varepsilon (u) = (u^2-1)(2u- \varepsilon \kappa )\), for \(\kappa \in {\mathbb R}\), as well as more general, but qualitatively similar, nonlinearities. We prove that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to \(\kappa \). In particular, in one dimension these interfaces behave like a relativistic point particle subject to constant acceleration.
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Notes
Our sign conventions for the unit normal, and hence the mean curvature, are described in Sect. 7.3, where we also review some basic properties of mean curvature. These sign conventions are such that the curve around which the solution in Sect. 2 concentrates, with the orientation we have implicitly chosen there, in fact has “mean curvature” equal to \(-\kappa \) rather than \(\kappa \).
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R. L. Jerrard was partially supported by the National Science and Engineering Research Council of Canada under operating Grant 261955. The authors are grateful to Kyle Thompson for pointing out an error in the statement of Theorem 3.2.
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Galvão-Sousa, B., Jerrard, R.L. Accelerating fronts in semilinear wave equations. Rend. Circ. Mat. Palermo 64, 117–148 (2015). https://doi.org/10.1007/s12215-014-0185-3
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DOI: https://doi.org/10.1007/s12215-014-0185-3