Introduction
Water wave analysis plays a pivotal role in varied domains such as coastal management and engineering, meteorological forecasting and tsunami prediction, renewable energy, the marine and shipping industries, and scientific research. Nonlinear partial differential equations (NLPDEs) are requisite for comprehending and forecasting the behavior of water waves under various situations. To address these equations, researchers have developed a powerful suite of mathematical techniques. For instance, methods such as the unified method [1] and the Jacobi elliptic function technique [2] have been instrumental in deriving periodic and quasi-periodic solutions, while expansion-based approaches, including the improved modified extended tanh function method [3] and the extended F-expansion method [4], are highly effective for extracting various traveling wave solutions. Other techniques, such as the extended simple equation method [5], the direct algebraic method [6], and the reduced differential transform method [7], have broadened the classes of obtainable analytical solutions. Recent advances in computational techniques, notably the bilinear neural network method [8] and neural network-based symbolic calculation approaches [9], have significantly augmented the methodological framework. These tools have been effectively employed to elucidate the nonlinear dynamics of cornerstone models like the Korteweg–de Vries (KdV) equation [10], Schrödinger equation [11], the Benjamin–Ono equation [12], the heat conduction equation [13], the Boussinesq equation [14], the Kadomtsev–Petviashvili (KP) equation [15], and the Kakutani-Matsuuchi (KM) equation [16], which govern complex wave regimes from shallow water to internal gravity waves.
Internal Gravity Waves (IGWs), a particularly important phenomenon governed by such equations, form in deep ocean layers, which arise from fluctuations in temperature and salinity that cause water stratification. These waves are fundamental to the distribution of momentum and energy and are essential for mixing processes within the ocean, influencing everything from underwater navigation and ocean current tracking to the nutrient cycles that support marine life [17]. Over the past two decades, observations have revealed large, high-amplitude IGWs appearing as solitary waves, solitons, or undular bores, often produced by tidal currents traversing uneven subaqueous topographies [18], [19], [20]. Theoretical research indicates that the evolution of these nonlinear IGWs is frequently governed by KdV-type equations, with the degree of nonlinearity and dispersion influenced by the surrounding equilibrium state [21], [22], [23], [24].
While traditional approaches often rely on irrotational inviscid fluid models [25], a more realistic understanding requires accounting for the influence of viscosity. This has led to the development of various models, including those for long gravity waves on viscous waters [26], [27], surface wave motion [28], and most notably for this study, the (1+1)-dimensional Kakutani-Matsuuchi (KM) model [16], [29]. As an extension of the KdV equation, the KM model, given by(1)integrates viscosity while preserving wave propagation.
In Eq. (1), describes the wave profile at position and time . The dispersive term accounts for third order dispersion effects on the wave, the time evolution term reflects changes in wave profile over time, the nonlinear term addresses nonlinear steepening and self-interaction of the wave, and the linear term pertains to the effects of linear dispersion. The KM Model provides advantages including nonlocal viscosity, nonlinearity, and analytical techniques for comprehending wave energy dissipation and profile development, while also addressing nonlinear elements such as energy transfer and wave steepening. The nonlocal viscous factor in the wave model elucidates the effects of viscosity on wave profiles, facilitating precise modeling of viscous water wave behavior and offering a thorough explanation of viscous influences.
The KM model has been a central focus of theoretical and analytical inquiry, aimed at understanding the intricate effects of viscosity on internal gravity waves. Foundational studies have mapped out some of its most critical behaviors. A key breakthrough was the detailed analysis of the model’s capacity for wave breaking, a phenomenon where solution slopes become infinite while amplitudes remain bounded [30]. This work underscored the model’s superior physical fidelity in a regime where many other viscous frameworks fail. Building on this understanding of its transient dynamics, subsequent research explored its long-term evolution, specifically the asymptotic stability and decay rates of its solutions. These studies clarified how wave energy dissipates over time due to the model’s characteristic nonlocal viscous term [31], [32]. The pursuit of exact analytical forms also yielded important results, as seen in the pioneering efforts by Khater and Alfalqi. Their work successfully applied methods like the Khater-II and auxiliary equation techniques to derive a foundational family of solitary wave structures [33].
Despite the significant insights provided by these seminal studies, their analytical scope has been largely focused on the foundational solitary wave structures derived from established techniques, leaving the full extent of the model’s richer soliton dynamics uncharted. The exploration of these soliton dynamics is essential, as such solutions are fundamental to understanding energy transport, hidden system characteristics, and complex wave interactions in fields ranging from fluid dynamics and optics to plasma physics [5], [34], [35], [36], [37]. Recognizing that the solutions reported in [33] represent only a fraction of the model’s capabilities, this study aims to significantly broaden the known solution space of the KM model. To address the deficiencies of previous approaches, we employ two powerful and contemporary analytical frameworks: the enhanced modified extended tanh expansion method (EMEThEM) [38] and the Sardar sub-equation approach [39]. EMEThEM represents a sophisticated extension of the tanh-function method, while the Sardar sub-equation approach leverages a quartic auxiliary equation to generate a wide array of solutions. Both frameworks are highly efficient for complex nonlinear systems where traditional perturbation techniques often fail. These methods are specifically chosen for their demonstrated capacity to yield a more diverse variety of wave structures, including complex W-shaped and singular solitons that often elude more traditional techniques. Beyond the discovery of new solutions, a primary contribution of this research is a detailed dynamical analysis of the KM model. We conduct a systematic investigation into the inherent stability of the system through bifurcation theory, identifying critical transition points, and explore its progression from ordered states to chaos under parametric forcing. This dual approach provides crucial, and to our knowledge, previously unavailable insights into the stability and complex behaviors of internal gravity waves.