Phase Portrait, Bifurcation and Chaotic Analysis, Variational Principle, Hamiltonian, Novel Solitary, and Periodic Wave Solutions of the New Extended Korteweg–de Vries–Type Equation

The center task of this paper is to give the qualitative and quantitative investigations into the nonlinear dynamics of the new extended Korteweg–de Vries–type equation for shallow-water waves. Applying the traveling wave transformation and semi-inverse method (SIM), the variational principle (VP) is developed. Based on the VP, we extract the system's Hamiltonian. The planar dynamical system is then derived using the Galilean transformation, followed by phase portrait plotting and bifurcation analysis to explore the existence of different types of wave solutions. Meanwhile, the chaotic behaviors of the system are also analyzed by taking the external perturbation terms. Eventually, two robust approaches—the variational method that stemmed from the variational principle and Ritz method—along with the Hamiltonian-based method are employed to seek some wave solutions of the equation. Different kinds of the wave solutions like bell shape solitary, anti-bell shape solitary, and periodic wave solutions are obtained. The findings of this exploration are all novel and help us gain a deeper understanding of the nonlinear dynamics of the equation being studied.