A novel variable-coefficient extended Davey–Stewartson system for internal waves in the presence of background flows

We propose a novel variable-coefficient Davey–Stewartson type system for studying internal wave phenomena in finite-depth stratified fluids with background flows, where the upper- and lower-layer fluids possess distinct velocity potentials, and the variable-coefficient terms are primarily controlled by the background flows. This realizes the first application of variable-coefficient DS-type equations in the field of internal waves. Compared to commonly used internal wave models, this system not only describes multiple types of internal waves, such as internal solitary waves, internal breathers, and internal rogue waves, but also aids in analyzing the impact of background flows on internal waves. We provide the influence of different background flow patterns on the dynamic behavior and spatial position of internal waves, which contribute to a deeper understanding of the mechanisms through which background flows influence internal waves. Furthermore, the system is capable of capturing variations in the velocity potentials of the upper and lower layers. We discover a connection between internal waves under the influence of background flows and velocity potentials. Through the variations in velocity potentials within the flow field, the dynamic behaviors of internal waves can be indirectly inferred, their amplitude positions located, and different types of internal waves distinguished. This result may help address the current shortcomings in satellite detection of internal wave dynamics and internal rogue waves.