DNA, or deoxyribonucleic acid, is found in every single cell and is the cell's primary information storage medium. DNA stores all an organism's genetic information, including the instructions it needs to grow, divide, and live. DNA is made up of four different building blocks called nucleotide bases: adenine (A), thymine (T), cytosine (C), and guanine (G). The genome is sequenced in vitro utilizing encoding strategies such as labelling one bond pair as 0 and the other as 1 to store digital information. In this study, the fractional differential order of double-chain DNA dynamical system was investigated, considering Atangana’s conformable fractional derivative. The conformable sub-equation method was applied to the system. The analysis resulted in some interesting new exact solutions of the model. One-soliton kink solution, multiple-soliton solutions, and periodic-wave solutions are the three broad categories that may be used to describe the results. In order to better understanding the solutions found, we have visually investigated a few of them. Both solitary and anti-solitary waves of the DNA strands are seen, attesting to the nonlinear dynamics of the system. The gathered data might be used to conduct application evaluations and draw further scientific findings.