This research examines the integrability and soliton solution in different forms of the Kairat-Ⅱ equation. A thorough Painlevé analysis determines that the equation fulfills the criteria of compatibility conditions for integrability; thus, it is shown to exhibit soliton solutions. Applying the Hirota D-operator, we transform the Kairat-Ⅱ equation into a trilinear form. This enables the derivation of precise analytical solutions, including novel breather wave and three-wave soliton solutions by the Hirota method. These solutions reveal localized, oscillatory, and energy-exchange structures, capturing the complex nonlinear dynamics of the model. We use a machine learning analysis, the multi-layer perceptron regressor algorithm, for understanding the dynamics of these solitons, which effectively predicts the progression and dynamics of the obtained solutions. The analytical results are validated and displayed in 2D and 3D+contour plots using symbolic computation tools such as Mathematica and Python. In the end, we execute the asymptotic analysis on the gained solution and present the asymptotic behavior of the solutions. We can learn a great deal about the intricate dynamics that the Kairat-Ⅱ model governs from these visualizations.

This paper proposes a novel hybrid fraud detection framework that integrates multi-stage feature selection, unsupervised clustering, and ensemble learning to improve classification performance in financial transaction monitoring systems. The framework is structured into three core layers: (1) feature selection using Recursive Feature Elimination (RFE), Principal Component Analysis (PCA), and Mutual Information (MI) to reduce dimensionality and enhance input relevance; (2) anomaly detection through unsupervised clustering using K-Means, Density-Based Spatial Clustering (DBSCAN), and Hierarchical Clustering to flag suspicious patterns in unlabeled data; and (3) final classification using a voting-based hybrid ensemble of Support Vector Machine (SVM), Random Forest (RF), and Gradient Boosting Classifier (GBC). The experimental evaluation is conducted on a synthetically generated dataset comprising one million financial transactions, with 5% labelled as fraudulent, simulating realistic fraud rates and behavioural features, including transaction time, origin, amount, and geo-location. The proposed model demonstrated a significant improvement over baseline classifiers, achieving an accuracy of 99%, a precision of 99%, a recall of 97%, and an F1-score of 99%. Compared to individual models, it yielded a 9% gain in overall detection accuracy. It reduced the false positive rate to below 3.5%, thereby minimising the operational costs associated with manually reviewing false alerts. The model’s interpretability is enhanced by the integration of Shapley Additive Explanations (SHAP) values for feature importance, supporting transparency and regulatory auditability. These results affirm the practical relevance of the proposed system for deployment in real-time fraud detection scenarios such as credit card transactions, mobile banking, and cross-border payments. The study also highlights future directions, including the deployment of lightweight models and the integration of multimodal data for scalable fraud analytics.

In this article, we investigated the integrability of the nonlinear dynamical Kairat-X model through Painlevé analysis, demonstrating that the equation satisfies the Painlevé property and is therefore integrable. We applied the bilinear Hirota method to derive several exact solutions, including breather wave, novel periodic wave, periodic cross-kink wave, kink-rogue wave interaction, and one-soliton and two-soliton solutions. A machine learning multi-layer-perceptron regressor algorithm was applied to represent the behavior of the actual, and to predict, the above solutions. Furthermore, we employed an asymptotic analysis on the gain solutions to expect the demonstration of the asymptotic behavior of these analytical solutions. The soliton solutions obtained were novel and exhibited improved reliability compared to previously reported results. These findings were further validated using symbolic computation software. A comparison with the existing literature revealed that the proposed solutions were more applicable and accurate. Several of the results were visualized using two-dimensional, three-dimensional, and contour surface plots.