We study the dynamics of domain walls in a double-field model in which the U(1) symmetry is broken both spontaneously and explicitly. The global U(1) symmetry of the system is restored when the symmetry breaking parameter λ is set to zero. Two pairs of degenerate kinks exist in the model, and they are related to each other by a Z2 transformation. We first calculate the single domain wall solutions and then investigate the collision processes. These include simple scattering, pair annihilation, pair capture, and other interesting processes. The possibility of the domain wall being punctured by a string is also investigated.

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