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Sine-Gordon model with variable mass (VMSG) appears in many physical systems,ranging from the current through nonuniform Josephson junction to DNA-promoterdynamics. Such models are usually nonintegrable with solutions foundnumerically or peturbatively. We construct a class of VMSG models, integrableboth at classical and quantum level with exact soliton solutions, which canaccelerate, change their shape, width and amplitude simulating realisticinhomogeneous systems at certain limits.Comment: 6 pages, 4 figures, revised with more physical input, to be published  in Phys. Rev. Let

Shape Changing and Accelerating Solitons in the Integrable Variable Mass Sine-Gordon Model