Toland, J. F. On a pseudo-differential equation for Stokes waves. (English) Zbl 1028.35126 Arch. Ration. Mech. Anal. 162, No. 2, 179-189 (2002). Author’s summary: It is shown that the existence of a smooth solution to a nonlinear pseudodifferential equation on the unit circle is equivalent to the existence of a globally injective conformal mapping in the complex plane which gives a smooth solution to the nonlinear elliptic free-boundary problem for Stokes waves in hydrodynamics.A dual formulation is used to show that the equation has no non-trivial smooth solutions, stable or otherwise, that would correspond to a Stokes wave with gravity acting in a direction opposite to that which is physically realistic. Reviewer: P.Godin (Bruxelles) Cited in 11 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35S05 Pseudodifferential operators as generalizations of partial differential operators 35R35 Free boundary problems for PDEs 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction Keywords:nonlinear pseudodifferential equation; conformal mapping; nonlinear elliptic free-boundary problem for Stokes waves PDFBibTeX XMLCite \textit{J. F. Toland}, Arch. Ration. Mech. Anal. 162, No. 2, 179--189 (2002; Zbl 1028.35126) Full Text: DOI