Global modes in nonlinear non-normal evolutionary models: exact solutions, perturbation theory, direct numerical simulation, and chaos

摘要

This paper is concerned with the theory of generic non-normal nonlinearevolutionary equations, with potential applications in Fluid Dynamics andOptics. Two theoretical models are presented. The first is a model two-levelnon-normal nonlinear system that not only highlights the phenomena of lineartransient growth, subcritical transition and global modes, but is also ofpotential interest in its own right in the field of nonlinear optics. Thesecond is the fairly familiar inhomogeneous nonlinear complex Ginzburg--Landau(CGL) equation. The two-level model is exactly solvable for the nonlinearglobal mode and its stability, while for the spatially-extended CGL equation,perturbative solutions for the global mode and its stability are presented,valid for inhomogeneities with arbitrary scales of spatial variation and globalmodes of small amplitude, corresponding to a scenario near criticality. Forother scenarios, a numerical iterative nonlinear eigenvalue technique ispreferred. Two global modes of different amplitudes are revealed in thenumerical approach. For both the two-level system and the nonlinear CGLequation, the analytical calculations are supplemented with direct numericalsimulation, thus showing the fate of unstable global modes. For the two-levelmodel this results in unbounded growth of the full nonlinear equations. For thespatially-extended CGL model in the subcritical regime, the global mode oflarger amplitude exhibits a `one-sided' instability leading to a chaoticdynamics, while the global mode of smaller amplitude is always unstable (theoryconfirms this). However, advection can stabilize the mode of larger amplitude.