  We are concerned with numerical approximations of breather solutions for the
cubic Whitham equation which arises as a water-wave model for interfacial
waves. The model combines strong nonlinearity with the non-local character of
the water-wave problem. The equation is non-integrable as suggested by the
inelastic interaction of solitary waves. As a non local model, it generalizes,
in the low frequency limit, the well known modified KdV (mKdV) equation which
is a completely-integrable model. The mKdV equation has breather solutions,
i.e. periodic in time and localized in space biparametric solutions. It was
recently shown that these breather solutions appear naturally as ground states
of invariant integrals, suggesting that such structures may also exist in
non-integrable models, at least in an approximate sense. In this work, we
present numerical evidence that in the non-integrable case of the cubic Whitham
equation, breather solutions may also exist.
