This study presents a numerical investigation of shallow water wave dynamics with particular emphasis on the role of surface tension. In the absence of surface tension, shallow water waves are primarily driven by gravity and are well described by the classical Boussinesq equation, which incorporates fourth-order dispersion. Under this framework, solitary and shock waves arise through the balance of nonlinearity and gravity-induced dispersion, producing waveforms whose propagation speed, amplitude, and width depend largely on depth and initial disturbance. The resulting dynamics are comparatively smoother, with solitary waves maintaining coherent structures and shock waves displaying gradual transitions. When surface tension is incorporated, however, the dynamics become significantly richer. Surface tension introduces additional sixth-order dispersive terms into the governing equation, extending the classical model to the sixth-order Boussinesq equation. This higher-order dispersion modifies the balance between nonlinearity and dispersion, leading to sharper solitary wave profiles, altered shock structures, and a stronger sensitivity of wave stability to parametric variations. Surface tension effects also change the scaling laws for wave amplitude and velocity, producing conditions where solitary waves can narrow while maintaining large amplitudes, or where shock fronts steepen more rapidly compared to the tension-free case. These differences highlight how capillary forces, though often neglected in macroscopic wave studies, play a fundamental role in shaping dynamics at smaller scales or in systems with strong fluid–interface interactions. The analysis in this work is carried out using the Laplace-Adomian Decomposition Method (LADM), chosen for its efficiency and accuracy in solving high-order nonlinear partial differential equations. The numerical scheme successfully recovers both solitary and shock wave solutions under the sixth-order model, with error analysis confirming remarkably low numerical deviations. These results underscore the robustness of the method while demonstrating the profound contrast between shallow water wave dynamics without and with surface tension.

This study presents a novel investigation into finite-time stability (FTS) and synchronization phenomena within a discrete reaction–diffusion system (RDS) governed by variable-order fractional (VOF) operators, inspired by the Degn–Harrison (D–H) model. By employing Caputo-type VOF differences, we model memory effects and time-varying dynamics typical of complex biological and chemical processes. Theoretical contributions include rigorous Lyapunov function (LF)-based criteria for establishing tempered Mittag-Leffler stability (MLS) and global FTS, as well as explicit expressions for the settling time $T^{\ast }$. A fractional-order (FO) error system is also analyzed, demonstrating that linear coupling ensures finite-time synchronization under variable-order conditions. Extensive numerical simulations confirm the theoretical predictions across various FO profiles $\delta (t)$ and parameter regimes. These findings bridge discrete fractional modeling with practical control strategies for systems exhibiting hereditary and anomalous diffusion effects.

Computational modeling plays a vital role in advancing our understanding and application of soliton theory. It allows researchers to both simulate and analyze complex soliton phenomena and discover new types of soliton solutions. In the present study, we computationally derive the bright and dark optical solitons for a Schrödinger equation that contains a specific type of nonlinearity. This nonlinearity in the model is the result of the combination of the parabolic law and the non-local law of self-phase modulation structures. The numerical simulation is accomplished through the application of an algorithm that integrates the classical Adomian method with the Laplace transform. The results obtained have not been previously reported for this type of nonlinearity. Additionally, for the purpose of comparison, the numerical examination has taken into account some scenarios with fixed parameter values. Notably, the numerical derivation of solitons without the assistance of an exact solution is an exceptional take-home lesson from this study. Furthermore, the proposed approach is demonstrated to possess optimal computational accuracy in the results presentation, which includes error tables and graphs. It is important to mention that the methodology employed in this study does not involve any form of linearization, discretization, or perturbation. Consequently, the physical nature of the problem to be solved remains unaltered, which is one of the main advantages.