  We use a one-scale similarity analysis which gives specific relations between
the velocity, amplitude and width of localized solutions of nonlinear
differential equations, whose exact solutions are generally difficult to
obtain. We also introduce kink-antikink compact solutions for the
nonlinear-nonlinear dispersion K(2,2) equation, and we construct a basis of
scaling functions similar with those used in the multiresolution analysis.
These approaches are useful in describing nonlinear structures and patterns, as
well as in the derivation of the time evolution of initial data for nonlinear
equations with finite wavelength soliton solutions.
