The fifth-order nonlinear water wave equation was explored in this and its utility in oceanography and its continued establishment was highlighted. Lie symmetry theory was applied to the nonlinear model, and the corresponding infinitesimal generators were constructed. Using the theory of abelian algebra and a suitable process of similarity reduction, the governing equation was simplified to a nonlinear ordinary differential equation. A new extended algebraic method, the nonlinear evolutionary differential approximation method, was presented to obtain the wave profiles by formulation of very general analytical solutions. To get a more detailed idea of how the physical processes that include nonlinear water waves work, 2D and 3D plots were created for several sets of parameter values, showing how the solitons were formed with unique shapes and the combined effects of dispersion and nonlinearly. In addition, a physics-informed neural network was deployed for the analysis of wave profiles. In this context, a 3D graphical representation was depicted with 2D training graphs. Additionally, by use of traveling wave transformation, 2D plots were presented to show how the wave profiles vary when parameter changes. The results, were obtained when the physics-informed neural network, were validated with a numerical scheme and revealed that the variation of parameters is crucially important to nonlinear oceanographic theories. These results support adequate parameter choices in the modeling of wave propagation and interaction in nonlinear water waves.
Citation: Hassan Almusawa, Zain Majeed, Adil Jhangeer. Application-oriented analysis of nonlinear water waves via analytical and neural network approaches; Oceanography Advances[J]. AIMS Mathematics, 2025, 10(11): 27635-27665. doi: 10.3934/math.20251215
The fifth-order nonlinear water wave equation was explored in this and its utility in oceanography and its continued establishment was highlighted. Lie symmetry theory was applied to the nonlinear model, and the corresponding infinitesimal generators were constructed. Using the theory of abelian algebra and a suitable process of similarity reduction, the governing equation was simplified to a nonlinear ordinary differential equation. A new extended algebraic method, the nonlinear evolutionary differential approximation method, was presented to obtain the wave profiles by formulation of very general analytical solutions. To get a more detailed idea of how the physical processes that include nonlinear water waves work, 2D and 3D plots were created for several sets of parameter values, showing how the solitons were formed with unique shapes and the combined effects of dispersion and nonlinearly. In addition, a physics-informed neural network was deployed for the analysis of wave profiles. In this context, a 3D graphical representation was depicted with 2D training graphs. Additionally, by use of traveling wave transformation, 2D plots were presented to show how the wave profiles vary when parameter changes. The results, were obtained when the physics-informed neural network, were validated with a numerical scheme and revealed that the variation of parameters is crucially important to nonlinear oceanographic theories. These results support adequate parameter choices in the modeling of wave propagation and interaction in nonlinear water waves.
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Hassan Almusawa, Zain Majeed, Adil Jhangeer. Application-oriented analysis of nonlinear water waves via analytical and neural network approaches; Oceanography Advances[J]. AIMS Mathematics, 2025, 10(11): 27635-27665. doi: 10.3934/math.20251215
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