Wide Localized Solitons in Systems with Time and Space-Modulated Nonlinearities

In this work we apply point canonical transformations to solve some classes of nonautonomous nonlinear Schr\"{o}dinger equation namely, those which possess specific cubic and quintic - time and space dependent - nonlinearities. In this way we generalize some procedures recently published which resort to an ansatz to the wavefunction and recover a time and space independent nonlinear equation which can be solved explicitly. The method applied here allow us to find wide localized (in space) soliton solutions to the nonautonomous nonlinear Schr\"{o}dinger equation, which were not presented before. We also generalize the external potential which traps the system and the nonlinearities terms.