EXISTENCE AND NONEXISTENCE OF SOLITARY WAVE SOLUTIONS TO HIGHER-ORDER MODEL EVOLUTION-EQUATIONS

The problem of existence of solitary wave solutions to some higher-order model evolution equations arising from water wave theory is discussed. A simple direct method for finding monotone solitary wave solutions is introduced, and by exhibiting explicit necessary and sufficient conditions, it is illustrated that a model admit exact sech2 solitary wave solutions. Moreover, it is proven that the only fifth-order perturbations of the Korteweg-deVries equation that admit solitary wave solutions reducing to the usual one-soliton solutions in the limit are those admitting families of explicit sech2 solutions.