The current paper retrieves optical 1–soliton solution to the concatenation model with the Kerr law of self-phase modulation structure. The primary mode of integration is the Lie symmetry analysis. The subsequent ordinary differential equations (ODEs) are formulated by the recently proposed Kumar–Malik scheme. This leads to the emergence of a variety of 1–soliton solutions along with their respective existence criteria. The numerical simulations support the analytical solutions that are yielded by the joint application of the two schemes.

The current paper retrieves implicit quiescent soliton solutions to optical metamaterials with nonlinear chromatic dispersion with generalized temporal evolution. Seven forms of self-phase modulation structures, as proposed by Kudryashov with time, are taken up. The implemented integration algorithm is Lie symmetry. A few of the solutions are in quadratures, while others are in terms of special functions. We also characterize the parameters that constrain the existence of such solutions.

The paper retrieves quiescent dispersive solitons in dispersion-flattened optical fibers having nonlinear chromatic dispersion and the Kerr law of self-phase modulation. The platform model is the Schrödinger–Hirota equation. The enhanced direct algebraic method has made this retrieval possible. The intermediary functions are Jacobi’s elliptic function and Weierstrass’ elliptic function. The final results appear with parameter constraints for the existence of such solitons.

This paper introduces a novel fractional‐order (FO) discrete‐time FitzHugh–Nagumo model (FHNM) that incorporates multiple time delays to capture the combined effects of memory and transmission latencies inherent in neuronal dynamics. The model is formulated using the Caputo nabla difference operator, providing a rigorous mathematical framework that preserves the hereditary properties of fractional calculus while enabling direct numerical simulation. We establish sufficient conditions for the uniqueness of solutions through contraction arguments and boundedness assumptions. Furthermore, by constructing an appropriate Lyapunov functional (LF), we derive a criterion for local Mittag–Leffler stability (MLS) of the equilibrium point (EP), a fractional generalization of exponential stability particularly suited for systems with long‐range memory. Comprehensive numerical simulations verify these theoretical predictions using realistic parameter values, demonstrating robust convergence to the EP across wide parameter ranges. An extensive sensitivity analysis explores the effects of FO, time delays, model parameters, and numerical implementation choices, identifying critical bifurcation thresholds where oscillations emerge via Hopf bifurcations. The results confirm the theoretical stability conditions and highlight the practical utility of the proposed framework for studying neuronal excitability under the influence of memory and delayed coupling. This work bridges the gap between continuous biophysical models and discrete data‐driven analyses, with potential applications in computational neuroscience, neurological disorder research, and neuromorphic computing.