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The novel nonlinear dispersive Gross-Pitaevskii (GP) mean-field model withthe space-modulated nonlinearity and potential (called GP(m, n) equation) isinvestigated in this paper. By using self-similar transformations and somepowerful methods, we obtain some families of novel envelope compacton-likesolutions spikon-like solutions to the GP(n, n) (n>1) equation. These solutionspossess abundant localized structures because of infinite choices of theself-similar function X(x). In particular, we choose X(x) as the Jacobiamplitude function am(x,k) and the combination of linear and trigonometricfunctions 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 andspikon solutions reported. Moreover, it is shown that GP(m,1) equation withlinear dispersion also admits the compacton-like solutions for the case 0<m<1and spikon-like solutions for the case m<0.

Localized Analytical Solutions and Parameters Analysis in the Nonlinear  Dispersive Gross-Pitaevskii Mean-Field GP (m,n) Model with Space-Modulated  Nonlinearity and Potential