University of Tabriz Computational Methods for Differential Equations 2345-3982 Articles in Press 2026 01 05 Optical solitons and bifurcation analysis in the cubic quintic septic nonic nonlinear Schrödinger equation using modified extended direct algebraic method 21072 10.22034/cmde.2025.68670.3342 EN Noshirvan Adil Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, Shenzhen University, Shenzhen 518060, China. Eman H. M. Abdullah Department of Physics and Engineering Math, The Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt. Hamdy M. Ahmed Department of Physics and Engineering Math, The Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt. Afaf A. S. Zaghrout Department of Mathematics, Faculty of Science, Al Azhar University, Cairo, Egypt. Youssra S. Ali Department of Mathematics, Faculty of Science, Al Azhar University, Cairo, Egypt. Wafaa B. Rabie Department of Mathematics, Faculty of Science, Luxor University, Taiba, Luxor, Egypt. Journal Article 2025 08 15 This study investigates the nonlinear Schr\"{o}dinger equation characterized by constant coefficients in a cubic–quintic–septic–nonic medium. By employing the modified extended direct algebraic method, a diverse range of analytical solutions is derived, including bright, dark, and mixed dark–bright soliton solutions, as well as singular structures. Additionally, the work presents periodic, exponential, rational, Jacobi elliptic, and Weierstrass elliptic solutions. To illustrate the physical characteristics of the obtained solutions, two-dimensional, three-dimensional, and contour plots are generated for specific parameter choices. A detailed bifurcation analysis is conducted to examine the system’s stability and dynamic transitions, supported by phase portraits and sensitivity analysis with respect to key parameters. https://cmde.tabrizu.ac.ir/article_21072_1b2f82c09f3419eb4e394338b08ea1ce.pdf