  In nonlinear dispersive evolution equations, the competing effects of
nonlinearity and dispersion make a number of interesting phenomena possible. In
the current work, the focus is on the numerical approximation of traveling-wave
solutions of such equations. We describe our efforts to write a dedicated
Python code which is able to compute traveling-wave solutions of nonlinear
dispersive equations of the general form \begin{equation*} u_t + [f(u)]_{x} +
\mathcal{L} u_x = 0, \end{equation*} where $\mathcal{L}$ is a self-adjoint
operator, and $f$ is a real-valued function with $f(0) = 0$.
  The SpectraVVave code uses a continuation method coupled with a spectral
projection to compute approximations of steady symmetric solutions of this
equation. The code is used in a number of situations to gain an understanding
of traveling-wave solutions. The first case is the Whitham equation, where
numerical evidence points to the conclusion that the main bifurcation branch
features three distinct points of interest, namely a turning point, a point of
stability inversion, and a terminal point which corresponds to a cusped wave.
  The second case is the so-called modified Benjamin-Ono equation where the
interaction of two solitary waves is investigated. It is found that is possible
for two solitary waves to interact in such a way that the smaller wave is
annihilated. The third case concerns the Benjamin equation which features two
competing dispersive operators. In this case, it is found that bifurcation
curves of periodic traveling-wave solutions may cross and connect high up on
the branch in the nonlinear regime.
