Exploring nonlinear equations and systems with predetermined coefficient values constrains the depth of understanding of the dynamics inherent in various applications and phenomena represented by such equations. On the contrary, exploration of nonlinear models with free coefficients offers avenues for improved development and ongoing refinement. In light of this, this study aimed to reassess the Wu–Zhang (WZ) system and Sawada–Kotera (SK) model by introducing arbitrary coefficients. Our goal is to identify the constraints necessary to ensure the existence of soliton solutions. Through the application of two distinct approaches, namely, the sine–cosine function method and tanh–coth expansion method, we systematically examine the conditions that facilitate the emergence of solitons within the WZ system and SK model. The insights gained from this analysis are supported by the presentation of 2D and 3D plots, providing a visual depiction of the propagation characteristics exhibited by the obtained solutions. The findings of the current work on conditions for the existence of soliton solutions for both generalized Wu–Zhang and generalized Sawada–Kotera models are novel and presented here for the first time.

Salt-and-pepper noise severely degrades the visual quality of digital images, par ticularly at high noise densities where conventional denoising techniques often fail. Median- and mean-based filters tend to oversmooth images and blur fine structures when the majority of pixels within a local window are corrupted. This paper proposes a robust dual-layer denoising framework for grayscale images that integrates rank-based prescreening, interquartile range (IQR)-based statis tical outlier detection using Tukey fences, and a lightweight post-processing sharpening stage. In the first layer, a rank-4 trimmed estimator suppresses ex treme impulse values and stabilizes local statistics. In the second layer, adap tive IQR thresholds are employed to detect and replace residual outliers, even in heavily corrupted neighborhoods. A final step involving selective sharpen ing combined with mild smoothing enhances edge details without amplifying residual noise. Extensive experiments on standard grayscale images (Lenna, Barbara, lake, boat, and living room) across salt-and-pepper noise levels from 10% to 90% demonstrate that the proposed approach consistently outperforms conventional methods, including mean, median, Gaussian, modified decision based unsymmetrical trimmed median filter (MDBUTMF), and pixel density based filter (BPDF). Quantitative evaluation indicates peak signal-to-noise ratio (PSNR) values reaching 38.23dB, structural similarity index (SSIM) values up to 0.99, and significant reductions in mean squared error (MSE), particularly at higher noise densities. These results confirm that the proposed framework ef fectively suppresses noise while preserving edges and textures, making it well suited for practical applications such as medical imaging, remote sensing, and surveillance.

The red palm weevil (Rhynchophorus ferrugineus, RPW) is one of the most destructive pests of date palms worldwide. Its larvae feed inside the stem, so damage is often irreversible before any symptoms are visible. Early detection and rapid removal of infested palms are therefore essential. We develop a five-compartment fractional-order mathematical model that tracks healthy palms, infested palms, larvae, adult weevils, and pheromone traps. The model uses fractional-order (Caputo) derivatives, which allow past infestation history to influence current dynamics. We use this memory as a phenomenological representation of delayed and cumulative effects, such as the 2–4 month larval development cycle, and we test it against integer-order, Erlang-distributed, and fixed-delay alternatives. We provide mathematical proofs that the model is well posed and identify a threshold, the basic reproduction number ${\mathcal{R}}_0$, that separates eradication from persistent infestation. The model is calibrated to two UAE datasets: the 1994/95 Al Ain survey of about $1.1\times {10}^6$ palms (${\mathcal{R}}_0$ > 1) and the 2025 national program outcomes (${\mathcal{R}}_0$ < 1), using the reported first and last counts of each series and a reconstructed monthly trajectory between them. This drop in ${\mathcal{R}}_0$ corresponds to three control improvements: sensor-based early detection, endotherapy (localized insecticide injection into the stem), and preventive stem injection. Model comparison supports a fractional order of about $0.95$ only for the 1994/95 epoch, whereas the 2025 data are described equally well without memory. Sensitivity analysis shows that faster removal of infested palms, lower RPW-to-palm contact rates, and reduced larval production are among the most effective control tactics. Model simulations of suspended and restored control suggest, as a hypothesis-generating scenario derived from hypothesized data, that a lapse in control can allow a transient resurgence, which supports continued monitoring. These results give quantitative guidance for planning pest management programs.