The stretch coordinate effect, bifurcation, and stability analysis of the nonlinear Hamiltonian amplitude equation

The new Hamiltonian amplitude equation effectively expresses the modulated wave instability and addresses the ill-posedness of the unstable nonlinear Schrödinger equation. This equation simulates nonlinear optical pulse propagation, fiber optic communication engineering, self-phase modulation, and modulated wave train instability. The unified tanh approach is used in this article to establish broad-spectral soliton solutions to the stated model in terms of hyperbolic and trigonometric functions. The solutions enfolded several free parameters associated with the model and the procedure, and specific values of these parameters result in some novel and typical soliton solutions that are examined in the texts. Additionally, the effect of the stretch coordinate ε is examined. The effects of stretching coordinates are determined by sketching three- and two-dimensional plots for different values of ε. The stability analysis of the gained solutions is examined, and the Hamiltonian function is discussed. Furthermore, we proceed to the bifurcation analysis of the model that is explored. To analyze the dynamic behavior of the solitons in nonlinear optics and other fields, the stability of the equilibrium points is evaluated, and a graphical representation of the system's phase diagram is provided.