On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

Comparability of Raman Spectroscopic Configurations: A Large Scale Cross-Laboratory Study

This is the final version. Available on open access from the American Chemical Society via the DOI in this recordThe variable configuration of Raman spectroscopic platforms is one of the major obstacles in establishing Raman spectroscopy as a valuable physicochemical method within real-world scenarios such as clinical diagnostics. For such real world applications like diagnostic classification, the models should ideally be usable to predict data from different setups. Whether it is done by training a rugged model with data from many setups or by a primary-replica strategy where models are developed on a 'primary' setup and the test data are generated on 'replicate' setups, this is only possible if the Raman spectra from different setups are consistent, reproducible, and comparable. However, Raman spectra can be highly sensitive to the measurement conditions, and they change from setup to setup even if the same samples are measured. Although increasingly recognized as an issue, the dependence of the Raman spectra on the instrumental configuration is far from being fully understood and great effort is needed to address the resulting spectral variations and to correct for them. To make the severity of the situation clear, we present a round robin experiment investigating the comparability of 35 Raman spectroscopic devices with different configurations in 15 institutes within seven European countries from the COST (European Cooperation in Science and Technology) action Raman4clinics. The experiment was developed in a fashion that allows various instrumental configurations ranging from highly confocal setups to fibre-optic based systems with different excitation wavelengths. We illustrate the spectral variations caused by the instrumental configurations from the perspectives of peak shifts, intensity variations, peak widths, and noise levels. We conclude this contribution with recommendations that may help to improve the inter-laboratory studies.COST (European Cooperation in Science and Technology)Portuguese Foundation for Science and TechnologyNational Research Fund of Luxembourg (FNR)China Scholarship Council (CSC)BOKU Core Facilities Multiscale ImagingDeutsche Forschungsgemeinschaft (DFG, German Research Foundation