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Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schr\"odinger equation
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For the Schr\"odinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the energy space. For this model, perturbative of the integrable cubic Schr\"odinger equation for small solutions, the linearized equation around a small solitary wave has an internal mode, whose contribution to the dynamics is handled by the Fermi golden rule.
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Kink dynamics for the Yang-Mills field in an extremal Reissner-Nordstr\"om black hole
For the Bizon-Kahl Yang-Mills kink in an extremal Reissner-Nordström background, globally bounded perturbations are shown to converge locally in space, and a finite-codimensional stable manifold is built.
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