We present analytical and numerical solutions for spatiotemporal chirped solitary waves in the generalized ( \(3+1\) )-dimensional nonlinear Schrödinger equation with varying sources under different modulated diffraction and potential functions. Our analytical approach provides a general formula that enables us to generate various types of solitary waves by introducing specific functional forms into the designed diffraction and potential profiles. We investigate the compressed, breathing, and quasiperiodic solitary wave solutions, discussing their characteristics and physical applications in relevant fields. Additionally, we explore numerical solutions where parameters can be arbitrarily chosen, encompassing both snakelike and J-shaped solitary waves. The results demonstrate that transverse and longitudinal structures of these solitary waves can be effectively manipulated by appropriately tuning the diffraction, potential strength, and source term. Furthermore, we numerically investigate solution stability by adding initial white noise; our simulations show that propagation shapes of the solitary waves are preserved.
