  Pseudospectral collocation methods and finite difference methods have been
used for approximating an important family of soliton like solutions of the
mKdV equation. These solutions present a structural instability which make
difficult to approximate their evolution in long time intervals with enough
accuracy. The standard numerical methods do not guarantee the convergence to
the proper solution of the initial value problem and often fail by approaching
solutions associated to different initial conditions. In this frame the
numerical schemes that preserve the discrete invariants related to some
conservation laws of this equation produce better results than the methods
which only take care of a high consistency order. Pseudospectral spatial
discretization appear as the most robust of the numerical methods, but finite
difference schemes are useful in order to analyze the rule played by the
conservation of the invariants in the convergence.
