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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年4月15日

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  • Fernando R. Humire,
  • Karin Alfaro-Bittner,
  • Marcel G. Clerc,
  • René G. Rojas

摘要

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.