Dynamics of the generalized $(3+1)$-dimensional nonlinear Schrödinger equation in cosmic plasmas

Under investigation in this paper is a generalized $(3+1)$-dimensional nonlinear Schrödinger equation with the variable coefficients, which governs the nonlinear dynamics of the ion-acoustic envelope solitons in the magnetized electron-positron-ion plasma with two-electron temperatures in space or astrophysics. Bilinear forms and Bäcklund transformations are derived through the Bell polynomials. $N$-soliton solutions are constructed in the form of the double Wronskian determinant and the $N$-th order polynomials in $N$ exponentials. Shape and motion of one soliton have been graphically analyzed, as well as the interactions of two and three solitons. When $\beta(t)$ and $\gamma(t)$ are both the periodic functions of the reduced time $t$, where $\gamma(t)$ is the loss (gain) coefficient, and $\beta(t)$ means the combined effects of the transverse perturbation and magnetic field, the shape and motion of one soliton as well as the interactions of two or three solitons will occur periodically. All the interactions can be elastic with certain coefficients.