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We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls–Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma–Balogh equations (Groma–Balogh in Acta Mater 47(13):3647–3654, 1999), a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of Patrizi and Sangsawang (Nonlinear Anal 202:112096, 2021). The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them. [ABSTRACT FROM AUTHOR]

We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls–Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a well known equation called by Head (1972) "the equation of motion of the dislocation continuum". The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan's law which states that dislocations move at a velocity proportional to the effective stress. [ABSTRACT FROM AUTHOR]

Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves ("isolas"), or the length increases to infinity so that branches are unbounded in function space ("snaking"). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyze the structure of branches of localized radial roll solutions in dimension 1+ε, with 0 < ε ≪ 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case. [ABSTRACT FROM AUTHOR]