In a world of sophisticated observing, measuring, and manufacturing methods, there is a growing need for suitable mathematical modeling tools. Partial differential equations (PDEs) are crucial for describing phenomena, such as water-wave movement or light propagation. Although PDEs have a long history, analyzing them remains among the most challenging fields of applied mathematics. This project seeks to refine analytical techniques that simplify model complexity by focusing on localized structures with heterogeneous tails. Such solutions are expected in relevant applications: solitary waves and wave packets for water wave equations, or pulse-like solutions for PDEs describing light flow in optical media.