Vladimir Kozlov (Linköping)
Stokes waves on rotational flows with counter-currents
The nonlinear problem under consideration describes two-dimensional, steady waves with vorticity on the free surface of water occupying a horizontal open channel of uniform rectangular cross-section. The aim is to investigate the bifurcation mechanism resulting in the formation of Stokes waves on the horizontal free surface of a shear flow in which counter-currents may be present. The whole family of these flows was studied in a joint paper with N. Kuznetsov [see QJMAM, 64 (2011)], in which, in particular, the expressions for their depths were derived. Here, the explicit conditions are presented that guarantee the existence of Stokes waves on a shear flow. It occurs that there are two different sets of conditions: one of these sets describes the case when the Bernoulli constant is fixed and a bifurcating parameter is related to the wavelength; on the contrary, the wavelength is fixed in the second case, whereas the Bernoulli constant varies. For unidirectional subcritical flows both types of conditions are always satisfied. General theorems are illustrated by several examples.
This is a joint work with N. Kuznetsov, Russian Academy of Sciences, St. Petersburg.