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Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen\u2013Cahn and the Fisher\u2013KPP equations on the manifold of fixed-rank matrices. Our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. 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