Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $H^{\frac12}(\mathbb{R}^2)$ is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein-Gordon equation in dimension $n=2$, which is captured by constructing an adapted systems of coordinate frames.

We prove global well-posedness and scattering for the massive Dirac-Klein-Gordon system with small initial data of subcritical regularity in dimension three. To achieve this, we impose a non-resonance condition on the masses.

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space H1(R3). The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein–Gordon equation. In a classical sense this fails and it is related to the failure of the endpoint Strichartz estimate for the wave equation in space dimension three. In this paper, systems of coordinate frames are constructed in which endpoint Strichartz estimates are recovered and energy estimates are established.

We consider the energy critical nonlinear Schrödinger equation on periodic domains of the form Rm × T4-m with m = 0,1, 2, 3. Assuming that a certain L4 Strichartz estimate holds for solutions to the corresponding linear Schrödinger equation, we prove that the nonlinear problem is locally well-posed in the energy space. Then we verify that the L4 estimate holds if m = 2, 3. © De Gruyter 2014.

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new wellposedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results are restricted to radially symmetric data.

The shape of a wave carries all information about the spatial and temporal structure of its source, given that the medium and its properties are known. Most modern imaging methods seek to utilize this nature of waves originating from Huygens' principle. We discuss the retrieval of the complete kinetic energy distribution from the acoustic trace that is recorded when a short ion bunch deposits its energy in water. This novel method, which we refer to as Ion-Bunch Energy Acoustic Tracing (I-BEAT), is a generalization of the ionoacoustic approach. Featuring compactness, simple operation, indestructibility and high dynamic ranges in energy and intensity, I-BEAT is a promising approach to meet the needs of petawatt-class laser-based ion accelerators. With its capability of completely monitoring a single, focused proton bunch with prompt readout it, is expected to have particular impact for experiments and applications using ultrashort ion bunches in high flux regimes. We demonstrate its functionality using it with two laser-driven ion sources for quantitative determination of the kinetic energy distribution of single, focused proton bunches.

The shape of a wave carries all information about the spatial and temporal structure of its source, given that the medium and its properties are known. Most modern imaging methods seek to utilize this nature of waves originating from Huygens' principle. We discuss the retrieval of the complete kinetic energy distribution from the acoustic trace that is recorded when a short ion bunch deposits its energy in water. This novel method, which we refer to as Ion-Bunch Energy Acoustic Tracing (I-BEAT), is a refinement of the ionoacoustic approach. With its capability of completely monitoring a single, focused proton bunch with prompt readout and high repetition rate, I-BEAT is a promising approach to meet future requirements of experiments and applications in the field of laser-based ion acceleration. We demonstrate its functionality at two laser-driven ion sources for quantitative online determination of the kinetic energy distribution in the focus of single proton bunches.