Abstract
The Kelvin–Helmholtz stability problem is studied by employing the variational method. With the restriction to a single dominant spatial mode, a set of fully nonlinear evolution equations is derived. The limiting states of these evolution equations are discussed and their stability is analyzed. It is found that at the critical point for the linear stability problem, another stable limiting state with finite amplitude appears. A nonlinear sinusoidal wave state is also found. The nonlinear dispersion relation for this wave is derived. The wavenumber of the dominant spatial mode and its evolution with time can also be determined with the aid of the variational method.
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