  A nonlinear Langmuir wave in the kinetic regime $k\lambda_D\gtrsim0.2$ may
have a filamentation instability, where $k$ is the wavenumber and $\lambda_D$
is the Debye length. The nonlinear stage of that instability develops into the
filamentation of Langmuir waves which in turn leads to the saturation of the
stimulated Raman scattering in laser-plasma interaction experiments. Here we
study the linear stage of the filamentation instability of the particular
family \cite{RoseRussellPOP2001} of Bernstein-Greene-Kruskal (BGK) modes
\cite{BernsteinGreeneKruskal1957} that is a bifurcation of the linear Langmuir
wave. Performing direct $2+2D$ Vlasov-Poisson simulations of collisionless
plasma we find the growth rates of oblique modes of the electric field as a
function of BGK's amplitude, wavenumber and the angle of the oblique mode's
wavevector relative to the BGK's wavevector. Simulation results are compared to
theoretical predictions.
