Dynamical Study of Newly Created Analytical Solutions, Bifurcation Analysis, and Chaotic Nature of the Complex Kraenkel-Manna-Merle System

The present research explores the analytical solutions and dynamics of complex Kraenkel-Manna-Merle system, which are exploited in ceramic-like materials with magnetic characteristics in electronics. Two distinct approaches, the extended sinh-Gordon equation expansion and the modified auxiliary equation are employed to derive soliton solutions in various function forms, including hyperbolic, trigonometric, rational, and Jacobi elliptic functions. The stability and accuracy of these solutions are confirmed through modulation instability analysis. Several specific solutions are illustrated through numerical simulations after assigning values to the free parameters. By back-substituting into the original model, the newly generated solutions confirm the validity of the new findings and ensure their accuracy. The obtained multiple soliton solutions show that the two approaches are effective, efficient, reliable, and potent for studying nonlinear evolution equations. Furthermore, a transformation converted the system into a planar dynamical system, allowing phase portraits to be analyzed. Moreover, the introduction of a perturbed term uncovered chaotic behavior across a range of parameter values, illustrated through both two-dimensional and three-dimensional graphics. The study presents novel analytical solutions that offer insights into nonlinear short wave interactions, which have not been previously documented using these methodologies.