This study examines the dynamical behavior of the dual-mode nonlinear Schrödinger equation (d-mNLSE), which describes the interaction, amplification, and attenuation of two coexisting wave modes in nonlinear media. The model incorporates key physical parameters including the nonlinearity coefficient, interaction phase velocity, and dispersion parameter, which significantly influence the evolution of nonlinear waves. By applying the modified Sardar sub-equation method (mSS-EM), a wide spectrum of exact analytical solutions is derived. These solutions include mixed trigonometric waves, shock-type structures, singular solutions, complex dark–bright solitons, multi-peak solitons, periodic and mixed-periodic waves, as well as mixed hyperbolic structures. The analytical findings provide useful insight into nonlinear wave propagation phenomena arising in fluid mechanics, water wave dynamics, ocean engineering, and related physical systems. Moreover, the conservation laws of the d-mNLSE are established, which leads to the conserved quantities of impulse power, momentum, and energy and describes the invariant characteristics of the soliton solutions during their propagation. The bifurcation analysis of the reduced dynamical model is carried out to explore the qualitative characteristics of the obtained solutions. The equilibrium points of the considered model are calculated, and their stability properties are analyzed systematically. To demonstrate the physical characteristics of the obtained solutions, different kinds of two-dimensional, three-dimensional, and contour plots are plotted using symbolic computations software. These findings confirm that the analytical method used to obtain the soliton solutions can be used to obtain a variety of soliton solutions of nonlinear evolution equations that appear in applied sciences and engineering.
Arshad et al. (Tue,) studied this question.