Suchergebnisse

Abstract. In this article we conduct a broad numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, a widely used model of rogue wave generation and dynamics in deep water. NLS breathers rising over an unstable background state are frequently used to model rogue waves. However, the issue of whether these solutions are robust with respect to the kind of random perturbations occurring in physical settings and laboratory experiments has just recently begun to be addressed. Numerical experiments for spatially periodic breathers with one or two modes involving large ensembles of perturbed initial data for six typical random perturbations suggest interesting conclusions. Breathers over an unstable background with N unstable modes are generally unstable to small perturbations in the initial data unless they are "maximal breathers" (i.e., they have N spatial modes). Additionally, among the maximal breathers with two spatial modes, the one of highest amplitude due to coalescence of the modes appears to be the most robust. The numerical observations support and extend to more realistic settings the results of our previous stability analysis, which we hope will provide a useful tool for identifying physically realizable wave forms in experimental and observational studies of rogue waves.

Abstract. Recently Gramstad and Trulsen derived a new higher order nonlinear Schrödinger (HONLS) equation which is Hamiltonian (Gramstad and Trulsen, 2011). We investigate the effects of dissipation on the development of rogue waves and downshifting by adding an additonal nonlinear damping term and a uniform linear damping term to this new HONLS equation. We find irreversible downshifting occurs when the nonlinear damping is the dominant damping effect. In particular, when only nonlinear damping is present, permanent downshifting occurs for all values of the nonlinear damping parameter β. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. We examine the generation of rogue waves in the presence of damping for sea states characterized by JONSWAP spectrum. Using the inverse spectral theory of the NLS equation, simulations of the NLS and damped HONLS equations using JONSWAP initial data consistently show that rogue wave events are well predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. We define δcutoff by requiring that 95% of the rogue waves occur for δ < δcutoff. We find that δcutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur. As a result when damping is present the proximity to homoclinic data and instabilities is more crucial for the development of rogue waves.

The toolbox for imaging molecules is well-equipped today. Some techniques visualize the geometrical structure, others the electron density or electron orbitals. Molecules are many-body systems for which the correlation between the constituents is decisive and the spatial and the momentum distribution of one electron depends on those of the other electrons and the nuclei. Such correlations have escaped direct observation by imaging techniques so far. Here, we implement an imaging scheme which visualizes correlations between electrons by coincident detection of the reaction fragments after high energy photofragmentation. With this technique, we examine the H2two-electron wave function in which electron-electron correlation beyond the mean-field level is prominent. We visualize the dependence of the wave function on the internuclear distance. High energy photoelectrons are shown to be a powerful tool for molecular imaging. Our study paves the way for future time resolved correlation imaging at FELs and laser based X-ray sources ; This work was funded by the Deutsche Forschungsgemeinschaft, the BMBF, the European Research Council under the European Union Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 290853 XCHEM, the MINECO projects FIS2013-42002-R and FIS2016-77889-R, and the European COST Action XLIC CM1204. All calculations were performed at the CCC-UAM and Mare Nostrum Supercomputer Centers. We are grateful to the staff of PETRA III for excellent support during the beam time. K.M. and M.M. would like to thank the DFG for support via SFB925/A3. A.K. and V.S. thank the Wilhelm und Else Heraeus-Foundation for support. J.L. would like to thank the DFG for support. S.K. acknowledges support from the European Cluster of Advanced Laser Light Sources (EUCALL) project which has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 654220. T.W. was supported by the U.S. Department of Energy Basic Energy Sciences under Contract No. DE-AC02-05CH11231. A.P. acknowledges a Ramón y Cajal contract from the Ministerio de Economa y Competitividad