Mathematical modelling and stability analysis for the thermal separation process of laser cutting will be considered. The process can be modelled as a free boundary problem of non-linear 2D partial differential equations with two moving boundaries describing the dynamical behaviour of the arising melt flow. An approximate reduced 1D model will be derived by asymptotic expansion methods. This model allows for linear stability analysis of the system, which will be applied for different levels of approximation qualities. The results are illustrated by numerical simulation of the free boundary and the reduced problem

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We experimentally show the existence of labyrinthine patterns between spatially modulated steady states in a simple nonlinear optical system consisting of a sodium vapor cell and a single feedback mirror. The inhomogeneity of the steady states leads to a locking phenomenon determining the length scale of the labyrinthine patterns and diminishes their region of existence in favor of localized periodic patterns. Numerical simulations confirm the experimental observations

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This work introduces a mathematical model for laser cutting taking account of spatially distributed laser radiation. The model involves two coupled nonlinear partial differential equations describing the interacting dynamical behaviors of the free boundaries of the melt during the process. The model will be investigated by linear stability analysis to study the occurence of ripple formations at the cutting surface. We define a measurement for the roughness of the cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the laser beam intensity along the free melt surface. Necessary optimality conditions will be deduced. Finally, a numerical solution will be presented and discussed by means of the necessary conditions. physical considerations