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We study one-and two-soliton solutions for the Cahn-Allen (CA) equation and the Brethorton equation. The CA equation has broad spectrum of applications especially in anti-phase boundary motion and it is used in phase-field models. While the Brethorton equation is a model for dispersive wave systems, it is used to find the resonant nonlinear interaction among three linear modes. We use the Hirota bilinear method to obtain one-and two-soliton solutions to the CA equation and the Brethorton equation.