Exploring two-dimensional internal waves: a new three-coupled Davey-Stewartson system and physics-informed neural networks with weight assignment methods (Q6191519)

The manuscript studies of various two-dimensional internal waves in stratified fluids using a newly derived \((2+1)\)-dimensional system of three coupled two-component Davey-Stewartson type equations. This system is capable of describing internal solitary waves, internal rogue waves, and internal breathers, highlighting the patterns of change in velocity potentials in the upper and lower layers as indicators for recognizing the occurrence, disappearance, and amplitudes of these internal waves. Furthermore, the manuscript introduces an improved physics-informed neural network (PINN) method for simulating the dynamics of these waves and analyzing the effects of different weight assignment methods in the loss function on training results. The scientific problem addressed in the paper is the investigation of various two-dimensional internal waves within a two-layer finite-water depth fluid, where different velocity potentials exist in each layer. The authors derived a novel \((2+1)\)-dimensional system of three coupled two-component Davey-Stewartson type equations under the long-wave assumption, which involves complex amplitude and real velocity potentials in both layers. To solve this problem, the authors employed the Hirota bilinear method to obtain exact solutions for internal solitary waves, internal rogue waves, and internal breathers along with their corresponding velocity potentials. These solutions demonstrate that the patterns of changes in velocity potentials can be used as indicators for recognizing the occurrence, disappearance, and amplitudes of these internal waves. Additionally, the manuscript introduces an improved PINN method, which successfully simulates the dynamics of these internal waves and their corresponding velocity potentials, providing a detailed analysis of the effects of no weight assignment and two different weight assignment methods in the loss function on the training results. The main findings of the study include the successful derivation of a novel \((2+1)\)-dimensional system capable of describing internal solitary waves, internal rogue waves, and internal breathers in stratified fluids. The study also found that the patterns of changes in the velocity potentials of the upper and lower layers can serve as indicators for recognizing the occurrence, disappearance, and amplitudes of these internal waves. Furthermore, the improved PINN method introduced in the study is effective in simulating the dynamics of these internal waves and their corresponding velocity potentials, with detailed analyses provided on the effects of different weight assignment methods in the loss function on the training results. The significance of this research lies in its contribution to the field of fluid dynamics and the study of internal waves in stratified fluids. The novel \((2+1)\)-dimensional system of three coupled two-component Davey-Stewartson type equations and the improved PINN method introduced in this paper offer new insights and tools for the simulation, prediction, and monitoring of internal waves. These advancements have the potential to impact the understanding of internal wave dynamics significantly, offering a valuable framework for future research in the area.