Abstract—In this study, the modified simplest equation method is employed to attain the exact solitary wave solutions to the nonlinear space-time fractional Phi-Four (PF) model incorporating the conformable derivative. The PF model is marked as a specific case of well-known Klein-Fock-Gordon model, abundant quantum consequence can be examined by PF model. The attained wave solutions are in types of trigonometric, rational, hyperbolic as well as their integration functions. This study introduces diverse structure of soliton like: multiple periodic and singular periodic soliton, bell and anti bell shape soliton, kink shape soliton and more. We also scrutinize the impact of fractional parameters on the behavior of wave circulation by generating surface plots of the solutions. The established results are realistic, wide-ranging, proficient and inclusive. This indicates that the employed technique is more efficient to resolve the different nonlinear models.

Keywords—the space-time fractional Phi-four model, the modified simplest equation method, the conformable derivative, soliton

Cite: M. Al-Amin, M. Mizanur Rahman, Mst. Kamrunnaher, M. Nurul Islam, M. Nazrul Islam, "Exploring the Nonlinear Behavior of Solitary Wave Structure to the Phi-Four Model for Quantum Consequence," International Journal of Applied Physics and Mathematics, vol. 16, no. 1, pp. 1-12, 2026.

Copyright © 2026 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).