In this work, we explore the dynamics of molecules in torsionally stressed DNA subjected to periodic external forces, specifically microwave radiation. Our approach involves constructing a novel continuum model based on a discrete model. Remarkably, this continuum model has not been analytically solved in existing literature, which motivates us to derive analytic solutions for investigating DNA s dynamical behavior. Our primary objective is to examine the impact of an external field (such as microwave radiation) on DNA dynamics, potentially affecting its structural integrity. Scientifically, we know that DNA molecules exposed to microwaves can suffer damage. Here, we focus on stability (or instability) to determine deterministic outcomes. Analytic solutions are essential for this purpose. The model equations governing torsional DNA (TDNA) behavior are non-autonomous and, in some cases, not integrable, meaning no exact solutions exist. Consequently, we rely on approximate solutions. Our chosen method is the extended unified method, allowing us to control errors through parameter selection. We consider two scenarios: when the torsional angle is smaller than one or completely free. Exact solutions emerge only when stacking and chain curvature constants are equal, otherwise, we derive approximate solutions. Numerical results: Numerical representations reveal that the localization of DNA molecules depends significantly on the microwave amplitude (MWA) and damping rate. Additionally, a critical MWA or DA value exists beyond which TDNA undergoes deformation. Stability analysis plays a crucial role in understanding these intricate dynamics. The present study sheds light on the interplay between external fields, DNA stability, and structural changes. Analytic solutions provide valuable insights into this complex system, with potential implications for biological processes and health.
