Nonlinear Partial Differential Equations in Fractal Spaces

Abstract

Nonlinear partial differential equations (NLPDEs) in fractal spaces create a strong mathematical structure which enables scientists to study wave propagation and soliton movements and strange particle movement across self-similar irregular shapes. The current paper presents a comprehensive study which includes multiple exact soliton solutions which researchers obtained through local fractional derivatives and M-truncated derivatives and advanced analytical methods which include the extended (G′/G)-expansion method and ϕ⁶-model expansion and Lie symmetry reduction and auxiliary equation approaches. We transform classical models which include the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) and Boussinesq and Kodama and cubic-quintic non-paraxial pulse equations into fractal spaces to obtain bright and dark and singular and periodic and combined and optical soliton molecules. The research paper provides a comprehensive examination of how stability testing and fractal order changes influence wave patterns. The results of this study support existing findings from 2025 which focus on nonlinear dispersive and optical systems while providing better understanding of surface geometry and shallow-water acoustics and optical fiber technologies in fractal materials.