New method is presented to look for exact solutions of nonlinear differentialequations. Two basic ideas are at the heart of our approach. One of them is touse the general solutions of the simplest nonlinear differential equations.Another idea is to take into consideration all possible singularities ofequation studied. Application of our approach to search exact solutions ofnonlinear differential equations is discussed in detail. The method is used tolook for exact solutions of the Kuramoto - Sivashinsky equation and theequation for description of nonlinear waves in a convective fluid. New exactsolitary and periodic waves of these equations are given.
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