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Sufficient conditions are given for stability of solitary-wave solutions of model equations for one-dimensional long nonlinear waves. These conditions differ from others which have appeared previously in that they are phrased in terms of positivity properties of the Fourier transforms of the solitary waves. Their use leads to simplified proofs of existing stability results for the Korteweg-de Vries, Benjamin-Ono, and Intermediate Long Wave equations; and to new stability results for certaindoi:10.1080/03605309208820831 fatcat:lyqb5ixntrfmxgwkp5lhoqlc7m