Abstract This study investigates solitary wave solutions of the three-dimensional, time-dependent nonlinear Schrödinger equation with cubic–quintic effects and a generalized Kudryashov-type self-phase modulation term. By applying the improved modified extended tanh function method, it obtains a broad spectrum of analytic solutions. These include bright soliton solutions, dark soliton solutions, singular periodic solutions, singular solutions, Jacobi elliptic function solutions, and a Weierstrass elliptic doubly periodic function solution. A detailed analysis demonstrates how variations in system parameters control the amplitude, width, and qualitative dynamics of the solitary waves. A comprehensive linear stability examination of the equilibrium points further reveals that parameter changes determine the emergence and disappearance of equilibrium states, with phase portraits illustrating the associated dynamical scenarios. The objectives are to establish the effectiveness of improved modified extended tanh function method in handling higher-dimensional nonlinear models, to enrich the catalogue of available exact solutions. The findings confirm that the proposed method is highly effective in generating exact solutions and in capturing intricate nonlinear structures influenced by the generalized Kudryashov contribution. This work expands the range of available solitary wave solutions and highlights their structural diversity. The study provides new theoretical insights into the interplay between nonlinearity, stability, and wave evolution, thereby offering valuable contributions to nonlinear optics, plasma physics, and higher-dimensional wave propagation.
Ahmed Tarek (2025) studied this question.