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Wolf-Patrick Düll NYU |
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| Phase diffusion in modulation equations for pattern forming systems | ||
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In order to describe slow modulations in time and space of spatially periodic solutions of pattern forming reaction-diffusion systems so-called phase diffusion equations and Cahn-Hilliard equations can be derived as formal approximation equations via multiple scaling analysis. If these phase diffusion equations are degenerate, there exist solutions showing a waiting time phenomenon. An example is the porous medium equation which can be derived as a degenerate phase diffusion equation for modulations of spatially periodic solutions of the real Ginzburg-Landau equation which have wave numbers close to the boundaries of the so-called Eckhaus-stable region. With the help of estimates between the formal approximations and the exact solutions of the original system we explain the extent to which these formal approximations are valid in different length and time scales. Furthermore, we show in which sense waiting time phenomena can occur in pattern forming systems. |