Localized structures close to translational symmetry-breaking point: noise-induced transitions to run-and-tumble motion

Macroscopic systems can exhibit particle-type solutions such as solitary waves, pulses, vegetation patches, and front solutions, among others. These localized structures may show active-like behavior when reflection symmetry is broken and fluctuations are present. Here, we investigate a one-dimensional non-variational system with multiplicative noise in which initially motionless structures drift and stochastically reverse their direction, resembling the run-and-tumble motion of bacteria. For low noise level intensities, the mean square displacement of the localized structure position displays a predominantly ballistic regimen with bimodal position distributions, while at high noise levels induce more frequent switching and complex transport dynamics, including initial diffusion, intermediate ballistic, and long-term diffusion regimes. A minimal model for the position and velocity captures run-and-tumble dynamics with rates governed by a Kramers law. The associated Fokker–Planck equation reproduces the full dynamics across all time scales, confirming the noise-induced transition in velocity and the diffusive behavior in position for long times. These findings show that active-like behavior can arise in nonlinear systems through both parity-breaking symmetry and noise.