Optical Soliton Solutions for Fractal Modified Zakharov–Kuznetsov Equation on Cantor Sets and Modulation Instability Analysis

In this paper, the new fractal modified Zakharov–Kuznetsov equation (fmZKe) defined on Cantor sets is investigated. The fmZKe is a non-differentiable model that arises naturally in mathematical physics, nonlinear wave theory, and plasma physics. The extended rational sine–cosine method is utilized to construct new optical soliton solutions of the model. The fmZKe is reduced to a non-differentiable ordinary differential equation by applying a non-differentiable wave transformation defined on Cantor sets. This reduction leads to a system of linear algebraic equations, which upon solving yields several exact solutions of the model. Furthermore, to establish a clear understanding of the model’s behavior, non-smooth graphical representations of the solutions are presented for various parameter values. The stability analysis of the newly obtained solutions in a classical sense is examined using stability theory, and the real-life applications of the results are highlighted.