Global bifurcation of steady gravity water waves with critical layers. (English) Zbl 1375.35294
Summary: We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann-Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler-Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.
MSC:
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 35R35 | Free boundary problems for PDEs |
| 35C07 | Traveling wave solutions |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35A15 | Variational methods applied to PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
Keywords:
gravity water waves; travelling water waves; Riemann-Hilbert problem; pseudodifferential equation; bifurcationReferences:
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