| Abstract: |
| We construct solitary waves for the two-dimensional steady water wave problem directly from the Babenko formulation in the case of pure gravity waves over finite depth. The proof is straight-forward in the sense that it is built on constrained minimization of a scalar nonlinear and nonlocal equation as in Weinstein (1987), and a limiting procedure from periodic waves as Ehrnstr\om-Groves-Wahl\`en (2012), using concentration--compactness. The novelty lies not the least in the nonlocal cubic terms appearing in the Babenko variational formulation, which carry negative order, and make nonlinear harmonic analysis an important part of the proof. The constructed smooth and localised solutions of the Babenko equation correspond to smooth and localised free-surface waves in the two-dimensional steady water wave problem over a flat bed with gravity. We show that the Babenko solitary waves are approximated by scalings of the classical Korteweg--De Vries solitary waves. |
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