Data-driven approach to shallow water equation in ocean engineering: Multi-soliton solutions, chaos, and sensitivity analysis

The objective of this research is to explore the dynamics of the shallow water wave equation in extended (3+1) dimensions. This equation is employed to represent atmospheric and oceanic turbulence from various standpoints. Utilizing a multiple exp-function technique, various solitary wave configurations in the form of 1-wave, 2-wave, and 3-wave are generated successfully. This approach is especially advantageous for extracting multisolitons without the need of bilinear forms. To visually illustrate and demonstrate the solutions, they are represented graphically using 3D, 2D, and density plots. Additionally, a qualitative nature of the dynamical system is conducted using bifurcation. Subsequently, an outward force is implemented to the model to create a disturbance, resulting in a modified planar system. The chaotic phenomenon in the modified system is confirmed through various tools designed for chaos detection. Further study is carried out on the model’s sensitivity under three different initial conditions, confirming that the system remains stable and does not exhibit high sensitivity. A newly introduced bidirectional scatter plot approach is employed to perform a comparative analysis of solution behaviors, effectively highlighting overlapping regions and distinctions within their solution spaces through data points, showcasing its innovative contribution. The results of this study are both intriguing and make a notable impact on the area of soliton specifically, as well as on the broader field of mathematical physics.