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


  • Zhenya Yan

    Corresponding author
    1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences
    • Address for correspondence: Zhenya Yan, Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, People's Republic of China; e-mail: zyyan@mmrc.iss.ac.cn

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The novel nonlinear dispersive Gross–Pitaevskii (GP) mean-field model with the space-modulated nonlinearity and potential (called GPmath formula 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 GPmath formula equation. These solutions possess abundant localized structures because of infinite choices of the self-similar function math formula. In particular, we choose math formula as the Jacobi amplitude function math formula 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 math formula and spikon-like solutions for the case math formula.