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Front propagation into unstable states
TL;DR: In this paper, the authors present an introductory review of the problem of front propagation into unstable states, which is centered around the concept of the asymptotic linear spreading velocity v ∗, the rate with which initially localized perturbations spread into an unstable state according to the linear dynamical equations obtained by linearizing the fully nonlinear equations about the unstable state.
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Symmetry reductions and exact solutions of a class of nonlinear heat equations
TL;DR: In this article, a catalogue of symmetry reductions for the nonlinear heat equation is given, including new reductions for linear heat equation and exact solutions of exact solutions for cubic heat equation for cubic f (u) in terms of the roots of f(u) = 0.
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The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation
TL;DR: In this paper, it was shown that there exist exact solutions of the Fitzhugh-Nagumo equation which can be obtained using the nonclassical method for determining symmetry reductions of partial differential equations developed by Bluman and Cole [J. Math. Phys. 18 (1969) 1025], but which are not obtained using a direct method as developed by Clarkson and Kruskal.
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Stationary non-equilibrium states of infinite harmonic systems
Herbert Spohn,Joel L. Lebowitz +1 more
TL;DR: In this paper, the existence, properties and approach to stationary non-equilibrium states of infinite harmonic crystals were investigated, and the authors obtained a product state, where the reservoirs are in equilibrium at temperatures β� −1¯¯¯¯ and β� − 1¯¯¯¯, and the system is in the unique stationary state of the reduced dynamics in the weak coupling limit.
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Spatial Waves of Advance with Bistable Dynamics: Cytoplasmic and Genetic Analogues of Allee Effects
TL;DR: Several new results are presented, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models and which has both basic and applied implications.