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Nonlinear evolution equations describe a variety of physical systems, at different scales from elementary particle models, to atomic and molecular physics, including fields like super-heavy nuclei, cluster radioactivity, atomic clusters, quantum hall drops, nonlinear optics, plasma and mesoscopic superconductor vortices, complex molecular systems, solid state, localized excited states, and Bose–Einstein condensates. At lab scale we have examples from fluid dynamics, pulses in nerves, swimming of motile cells and electric lines. Larger scale applications are related to tides in neutron stars or impact of stellar objects. It is of particular interest to examine the dynamics of localized solutions on compact domain of definitions like closed segments, closed curves, or closed surfaces, in one word on the boundaries of some compact domains.