Dark solitons, modulation instability and breathers in a chain of weakly non-linear oscillators with cyclic symmetry

摘要

In the aerospace industry the trend for light-weight structures and theresulting complex dynamic behaviours currently challenge vibration engineers.In many cases, these light-weight structures deviate from linear behaviour, andcomplex nonlinear phenomena can be expected. We consider a cyclically symmetricsystem of coupled weakly nonlinear undamped oscillators that could beconsidered a minimal model for different cyclic and symmetric aerospacestructures experiencing large deformations. The focus is on localisedvibrations that arise from wave envelope modulation of travelling waves. Forthe defocussing parameter range of the approximative nonlinear evolutionequation, we show the possible existence of dark solitons and discuss theircharacteristics. For the focussing parameter range, we characterise modulationinstability and illustrate corresponding nonlinear breather dynamics.Furthermore, we show that for stronger nonlinearity or randomness in initialconditions, transient breather-type dynamics and decay into bright solitonsappear. The findings suggest that significant vibration localisation may arisedue to mechanisms of nonlinear modulation dynamics.