Advanced study of nonlinear partial differential equations in ion-acoustic and light pulses: solitons, chaos, and energy perspectives

The emphasis of this work is to apply the complete discriminant system (CDS) of a polynomial method (CDSPM) to obtain soliton solutions for the extended Korteweg–de Vries equation (EKdVE). The model of the KdV equation is suitable for describing water waves with shallow water, where the water depth is up to the wavelength. We explore exact and analytical solutions such as the hyperbolic function (HF), the Jacobian elliptic function (JEF), solitary wave (SW) solutions, and trigonometric solutions. Additionally, we apply bifurcation concept in qualitative model of the EKdVE. The procedure involves transforming the system into a planer dynamical system through a given transformation and examining bifurcation analysis. We also explore phase portraits which assist in determining the stability of the equilibrium point of the system. Additionally, chaotic behaviour (CB) exhibits the exponential divergence of the orbits, i.e., tiny differences in the initial condition result in widely divergent orbits over a time period. To explore the potential of CB, we insert a perturbed term to the dynamical system and proceed with caution to explore the model. From these, we obtain a linear combination resulting in a strong dynamic mathematical structure. Finally, we apply Jupyter as a machine learning program as well as Mathematica to graph some obtained solutions in various dimensions such as three-dimensional (3D) and two-dimensional (2D) plots by using some packages of PYTHON such as numpy, scipy.integrate, and matplotlib.pyplot. Additionally, we explore energy balance method (EBM), presenting approximate periodic solution of non-linear oscillatory systems. The method is based on principle of conservation of energy, total energy (sum of kinetic energy as well as potential energy) is conserved.