An Invariant Optical Soliton Wave Study on Integrable Model: A Riccati-Bernoulli Sub-Optimal Differential Equation Approach

The double-chain deoxyribonucleic acid model, which is important to the retention and transfer of genetic material in biological domains, is examined in this study. It is important because, it bridges the gap between theoretical physics and molecular biology by offering a more thorough and precise explanation of DNA behavior. In this model, the bottom combination represents hydrogen bonds between base pairs in the two long, evenly elastic filaments that represent the two polynucleotide chains of the deoxyribonucleic acid molecule. The Lie symmetry analysis is used to explain the Lie invariance criteria. This leads to a four-dimensional Lie algebra where the translation point symmetries in space and time correlate with the conservation of mass and energy, respectively, and the remaining point symmetries are dilation and scaling. The double-chain deoxyribonucleic acid partial differential model is reduced to an ordinary differential equation, and built Lie subalgebras are found first, along with invariant closed-form solutions. The Cauchy problem for the double-chain deoxyribonucleic acid model cannot be solved by the inverse scattering transform method; therefore, the analytical Riccati-Bernoulli suboptimal differential equation approach technique is used to build the exact solution. The appropriate parametric values are taken in contour, two, and three dimensions to graphically illustrate the solution. A physically meaningful and intuitive interpretation of the system dynamics is required in order to make the Hamiltonian function under consideration easier to comprehend and analyze. One of the numerous conservation principles commonly seen in systems defined by a Hamiltonian function is energy conservation. The conservation laws are determined for the model under consideration, which are essential for deciphering and solving complex problems and are used to illustrate deep understandings of how physical systems behave. Understanding the stability and long-term behavior of the system depends on these preserved quantities. To assess the governing system's sensitivity, a sensitive analysis is offered.