×

Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. (English) Zbl 0755.76023

Summary: 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 admits exact \(\text{sech}^ 2\) solitary wave solutions. Moreover, it is proven that the only fifth- order perturbations of the Korteweg-de Vries 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.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35B25 Singular perturbations in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI Link