Novel analytical approaches and stability examination for soliton solutions in a dispersive perturbed Gardner model

This study presented a rigorous analytical investigation of the perturbed Gardner equation, a fundamental model describing nonlinear wave propagation in plasma physics, fluid dynamics, and nonlinear optics. Using an advanced analytical framework, we derived exact solutions capturing diverse wave behaviors: Localized bright solitons representing stable energy packets, dark solitons manifesting as intensity voids, singular solitons with distinctive phase profiles, and periodic waves governed by elliptic functions. Linear stability analysis revealed that solution robustness emerged from a delicate balance between nonlinear focusing and dispersive spreading, with precise parameter windows identified for maintaining structural integrity. A key finding identified the dispersion triplet—higher-order dispersion coefficients—as crucial for controlling wave amplitude, spectral broadening, and nonlinearity-dispersion balance. Numerical validation confirmed mathematical consistency and physical feasibility, with strong agreement between theory and simulations. This established a framework for understanding nonlinear wave phenomena in photonic and plasma applications.