Exact and Data-Driven Lump Wave Solutions for the (3+1)-Dimensional Hirota–Satsuma–Ito-like Equation

In this paper, the lump wave solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like (HSIl) equation are constructed by employing the Hirota bilinear method and quadratic function approach, and the corresponding propagation behaviors and nonlinear dynamical properties are also investigated. At the same time, the physics informed neural network (PINN) deep learning technique is employed to study the data-driven solutions for the HSIl equation from the derived lump wave solutions. The machine learning results show high effectiveness and accuracy, providing new techniques for discussing more nonlinear dynamics of lump waves and discovering new lump wave solutions.