From the solutions to construct the Schrödinger-like equation with source term and its numerical simulations.

We present a brand new method to construct the Schrödinger-like equations from a solution in this paper. Some Schrödinger-like equations with sources are derived by using a generalized solution, which have the form $$i\frac{\partial \Psi }{\partial x}+\frac{1}{2}\frac{\partial ^2\Psi }{\partial t^2}+|\Psi |^2\Psi +\alpha (x,t)\Psi +\beta (x,t)e^{ipx}=0$$ . The abundant analytical solutions of the Schrödinger-like equations are considered, including the bright rogue wave solution, dark rogue wave solution, Bell-shaped soliton solution, the interactions of two solitons, and other special soliton solutions. And we prove that the equation with source term has a weak solution. At last, the numerical simulations on the evolution and solitons collision of rogue wave solutions are performed to verify the prediction of the analytical formulations. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids. [ABSTRACT FROM AUTHOR]