Existence and Nonexistence of Solitary Wave Solutions to Higher-Order Model Evolution Equations

Abstract

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 ${\text{sech}}^2 $ 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 ${\text{sech}}^2 $ solutions.