Kichenassamy, Satyanad; Olver, Peter J. Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. (English) Zbl 0755.76023 SIAM J. Math. Anal. 23, No. 5, 1141-1166 (1992). 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. Cited in 96 Documents 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 Keywords:monotone solitary wave solutions; sech(square) solitary wave solutions; fifth-order perturbations; Korteweg-de Vries equation PDFBibTeX XMLCite \textit{S. Kichenassamy} and \textit{P. J. Olver}, SIAM J. Math. Anal. 23, No. 5, 1141--1166 (1992; Zbl 0755.76023) Full Text: DOI Link