  The novel nonlinear dispersive Gross-Pitaevskii (GP) mean-field model with
the space-modulated nonlinearity and potential (called GP(m, n) equation) is
investigated in this paper. By using self-similar transformations and some
powerful methods, we obtain some families of novel envelope compacton-like
solutions spikon-like solutions to the GP(n, n) (n>1) equation. These solutions
possess abundant localized structures because of infinite choices of the
self-similar function X(x). In particular, we choose X(x) as the Jacobi
amplitude function am(x,k) and the combination of linear and trigonometric
functions of space x so that the novel localized structures of the GP(2,2)
equation are illustrated, which are much different from the usual compacton and
spikon solutions reported. Moreover, it is shown that GP(m,1) equation with
linear dispersion also admits the compacton-like solutions for the case 0<m<1
and spikon-like solutions for the case m<0.
