Abstract:
We consider steady water waves in a two-dimensional channel. The water
motion is rotational with constant vorticity. We consider an analytic branch of
Stokes waves started from a subcritical laminar flow, where the period is
considered as the bifurcation parameter. The first eigenvalue of the Frechet
derivative on this branch is always negative. The main object of our study is the
second eigenvalue of the Frechet derivative at this branch in a neighborhood of
the laminar flow. This is a small eigenvalue, and the positive sign corresponds
to the confirmation of the principle of exchange of stability and the negative
sign to its violation. We consider the dependence of the sign on the depth of
the laminar flow and the value of the constant vorticity. We discuss the
connection of the sign of the second eigenvalue with subharmonic
bifurcations. We also verify the property of formal stability by a description of
the domain of the parameters of the problem, where this property holds.
This is joint work with Oleg Motygin (Institute for Problems in Mechanical
Engineering, Russian Academy of Sciences, St Petersburg).
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