A novel dynamic model and the oblique interaction for ocean internal solitary waves

Under investigation in this article is the propagation of the internal solitary waves in the finite depth ocean. According to the perturbation and multi-scale analysis method, the (2+1)-dimensional Kadomtsev–Patviashvili-intermediate long-wave (KP-ILW) equation is derived. This is a novel model for describing the ocean internal solitary waves for the first time. It should be noted that when $$a_2=0$$ , the model is converted to the ILW equation; when $$h_1-h_0\rightarrow 0$$ , the model is converted to the KP equation; when $$h_1\rightarrow \infty $$ , the model is changed to the KP-BO equation. In order to further study the properties of the internal solitary waves, we explore the conservation of momentum, mass and energy of the internal solitary waves. Through the Hirota bilinear method, we obtain the Bäcklund transformation of the KP-ILW equation for the first time. This is of great significance to the construction of the infinite conservation law of the model. Then, the N-soliton solutions of the KP-ILW equation are given. Meanwhile, we study the oblique interaction of the internal solitary waves, which leads to the discovery of rogue waves and Mach reflection. In addition, we also discuss the effect of some parameters on Mach stem and get some conclusions. All these are particularly important for studies of large amplitude waves such as tsunamis in shallow water.