Length-scale competition in the nonlinear Dirac equation under spatially periodic potential

We investigate the dynamics of a solitary wave in the nonlinear Dirac equation under a spatially periodic potential with wave number k. Using a two-collective-coordinate ansatz, we derive an effective equation of motion for the solitary wave’s center of mass based on energy conservation and on momentum continuity equation with a source term. The solitary wave behaves as a relativistic particle in an effective potential, which in the non-relativistic limit reduces to a nonlinear pendulum in a sinusoidal potential of period λ = 2π/k. The dynamics is governed by the interplay between the potential period λ and the solitary wave width Ls. The wave number k controls both the period and amplitude of the effective potential, leading to distinct regimes. For λ ≫ Ls, the solitary wave exhibits either bounded oscillations or unbounded motion, depending on its initial velocity; in contrast, for λ ≪ Ls, the potential amplitude is effectively negligible and the motion becomes uniform. In the particle-like regime, a critical initial velocity separates trapped and unbounded motion. At low frequencies, where the solitary wave develops a two-hump structure, we observe a λ/2 phase shift in the effective potential and the conversion of energy associated with internal shape oscillations into translational kinetic energy, a mechanism that lies beyond the point-particle description and reveals behavior not typically associated with length-scale competition in solitary wave systems.