Data-driven rogue waves solutions for the focusing and variable coefficient nonlinear Schrödinger equations via deep learning

In this paper, we investigate the data-driven rogue waves solutions of the focusing and the variable coefficient nonlinear Schrödinger (NLS) equations by the deep learning method from initial and boundary conditions. Specifically, first- and second-order rogue wave solutions for the focusing NLS equation and three deformed rogue wave solutions for the variable coefficient NLS equation are solved using physics-informed memory networks (PIMNs). The effects of optimization algorithm, network structure, and mesh size on the solution accuracy are discussed. Numerical experiments clearly demonstrate that the PIMNs can capture the nonlinear features of rogue waves solutions very well. This is of great significance for revealing the dynamical behavior of the rogue waves solutions and advancing the application of deep learning in the field of solving partial differential equations.