Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. I. Adjoint method

Isotropic chemically active particles can spontaneously self-propel owing to a symmetry-breaking instability when diffusion of the chemical is sufficiently weak relative to advection. This series revisits the weakly nonlinear theory describing the dynamics of such spontaneous swimmers near the instability threshold. Part I identifies the adjoint linearized equations, a key element towards a versatile theory that can be used to model general motion in three dimensions including perturbation effects and interactions. The general theoretical framework is demonstrated by deriving and analyzing steady amplitude equations in a range of physical scenarios.