Study of bifurcations, chaotic structures with sensitivity analysis and novel soliton solutions of non-linear dynamical model

The current work amalgamates the fascinating exploration of the Radhakrishnan–Kundu–Lakshmanan equation, which is a sophisticated mathematical framework in physics and finds applications in elucidating diverse physical phenomena. By leveraging the two recently developed computational approaches, namely the new extended direct algebraic method and the modified Sardar sub-equation method, we rigorously assess the novel soliton solutions including dark, bright-dark, dark-bright, periodic, singular, rational and mixed trigonometric forms. Furthermore, we also segregated W-shape, M-shape, bell shape, exponential, as well as hyperbolic soliton, which are not documented in the literature. We validate the stability and accuracy of extracted soliton wave solutions using the Hamiltonian property. Additionally, the Galilean transformation is applied and numerous standard types of results, including bifurcations, chaotic flows, and sensitivity analysis are presented. The obtained results are tested both numerically and with illuminating physical interpretations, which shows a better demonstration of the intricate dynamics of these models.