We show decay estimates for the propagator of the discrete Schrödinger and Klein-Gordon equations in the formThis implies a corresponding (restricted) set of Strichartz estimates. Applications of the latter include the existence of excitation thresholds for certain regimes of the parameters and the decay of small initial data for relevant l p norms. The analytical decay estimates are corroborated with numerical results.

We study the question of well-posedness of the Cauchy problem for Schrödinger maps from R 1 × R 2 to the sphere S 2 or to H 2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well-posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well-posedness of the Schrödinger map itself from it. In proving well-posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L 2 -functions.

We construct a gauge theoretic change of variables for the wave map from R × R n into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equationn ≥ 4 -for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4.

Convolution type Calderón-Zygmund singular integral operators with rough kernels p.v. Ω(x)/|x| n are studied. A condition on Ω implying that the corresponding singular integrals and maximal singular integrals map L p → L p for 1 < p < ∞ is obtained. This condition is shown to be different from the condition Ω ∈ H 1 (S n−1 ).

We consider the question of existence of "bell-shaped" (i.e., nonincreasing for x > 0 and nondecreasing for x < 0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p = 3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego (Proc. Am. Math. Soc. 133:1763, 2005 formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke-Wattis result (Commun. Math. Phys. 161:391, 1994) by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.