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We show that the leading semiclassical behavior of soliton form factors at arbitrary momentum transfer is controlled by solutions to a new wave-like integro-differential equation that describes solitons undergoing acceleration. We work in the context of two-dimensional linear sigma models for concreteness, but our methods are purely semiclassical and generalizable.  This talk is based on work done in collaboration with Ilarion Melnikov and Costis Papageorgakis.