Periodically driven systems provide unique opportunities to investigate the dynamics of topological excitations far from equilibrium. In this paper, we report a time-crystalline soliton (TCS) state in a driven-dissipative ${\varphi }^4$ model. This state exhibits spontaneous breaking of discrete time-translational symmetry while simultaneously displaying spatial soliton behavior. During time evolution, the soliton pattern periodically oscillates between kink and antikink configurations. We further study TCS dynamics under noise, demonstrating that soliton random walk can induce a dynamical transition between two distinct $Z_2$ symmetry-breaking time-crystalline phases in time domain. Finally, we examine the annihilation of two spatially separated TCSs under noise. Importantly, in contrast to the confined behavior of time-crystalline monopoles reported in Yue and Cai [Phys. Rev. Lett. 131, 056502 (2023)], the dynamics of time-crystalline solitons is deconfined despite the nonequilibrium nature of our model: the statistically averaged annihilation time scales as a power law with the solitons' initial separation.

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• Exotic phases of matter
• Landau theory
• Solitons
• Time crystals
• Topological defects
• Floquet systems
• Nonequilibrium systems
• Brownian dynamics
• Discrete symmetries in condensed matter
• Landau-Ginzburg theory
• Langevin algorithm
• Langevin equation
• Stochastic differential equations
• Time-dependent Ginzburg-Landau theory