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- ItemUniform asymptotic expansions for the infinite harmonic chain(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Patz, CarstenWe study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by 1/t uniformly in space. In particalur we give precise asymptotics for the transition from the 1/t1/2 decay of nondegenerate wave numbers to the generate 1/t1/3 decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.
- ItemDispersive stability of infinite dimensional Hamiltonian systems on lattices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Patz, CarstenWe derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein-Gordon chain. If the nonlinearity is of degree higher than 4, then small localized initial data decay like in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different $ell^p$ spaces