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

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 functionam ( 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 .