Self-organized Hydrodynamics in an Annular Domain: Modal Analysis and Nonlinear Effects

The Self-Organized Hydrodynamics model of collective behavior is studied on an annular domain. A modal analysis of the linearized model around a perfectly polarized steady-state is conducted. It shows that the model has only pure imaginary modes in countable number and is hence stable. Numerical computations of the low-order modes are provided. The fully non-linear model is numerically solved and nonlinear mode-coupling is then analyzed. Finally, the efficiency of the modal decomposition to analyze the complex features of the nonlinear model is demonstrated. Acknowledgements: This work was supported by the ANR contract ’MOTIMO’ (ANR11-MONU-009-01). The first author is on leave from CNRS, Institut de Mathématiques, Toulouse, France. He acknowledges support from the Royal Society and the Wolfson foundation through a Royal Society Wolfson Research Merit Award and by NSF Grant RNMS11-07444 (KI-Net). The second authors wishes to acknowledge the hospitality of the Department of Mathematics, Imperial College London, where this research was conducted. Both authors wish to thank F. Plouraboué (IMFT, Toulouse, France) for enlighting discussions.