Unveiling the Dynamics of (3+1) Dimensional <i>q</i> ‐Deformed Tanh‐Gordon Equation: A Controlled Picard Approach to Fractional Nonlinear Systems

ABSTRACT In this study, we delve into the intricate dynamics of the ( ) dimensional ‐deformed tanh‐Gordon equation, exploring its fractional form through the lens of the Caputo fractional derivative. By employing the innovative controlled Picard technique combined with Laplace transforms, we derive highly accurate approximate solutions, ensuring both convergence and reliability in the realm of fractional nonlinear equations. Our rigorous analysis of the solution's existence and uniqueness provides a robust theoretical foundation, while two‐ and three‐dimensional graphical representations vividly illustrate the profound impact of fractional and ‐deformed parameters on the system's behavior. The findings not only highlight the efficacy of the controlled Picard technique in tackling complex fractional models but also offer valuable insights into their numerical treatment. This research opens new avenues for exploring higher dimensional and coupled fractional systems, paving the way for future advancements in nonlinear dynamics and fractional calculus.