Pattern identification in systems with S(1) symmetry.

This work is devoted to pattern identification in systems with S(1) symmetry based on limited experimental data. As we demonstrate, such pattern identification is complicated by the lack of a theoretical basis as well as by the presence of experimental uncertainties, and possible overlapping and missing points in the data. The study is motivated by a recent finding of physical systems where instabilities of different wave numbers may coexist and thus lead to several single-wave-number patterns superimposed with a random phase-shift between them. As shown in this work, such patterns cannot be identified with Fourier analysis as well as direct measurement of the wave numbers is not possible. We present both a constructive theoretical approach, which establishes the conditions under which the structure of such patterns is identifiable, and an example of application-the crown structure analysis in the drop splash problem. For the latter study, an experimental setup is developed based on high-speed stereo photography, which produces data suitable for a quantitative analysis of the observed patterns.