Neural Network Modeling of Optical Solitons Described by Generalized Nonlinear Schrödinger Equations

The paper examines the modeling of pulse propagation in a nonlinear medium using two partial differential equations, namely the second-order Schrödinger equation and the fourth-order generalized nonlinear Schrödinger equation (GNSE). The applicability of physics-informed neural networks (PINN) methods for solving the GNSE is demonstrated for analyzing physical effects involving solitons, using the example of soliton interaction with an isolated wave. A study of the efficiency of balancing methods for the GNSE is conducted on a boundary value problem with zero boundary conditions, as well as an assessment of the accuracy of the PINN method with segmentation by comparing it to the exact solution for single solitons of the second and fourth-order GNSE. The use of conservation laws as a means of verifying the validity of the solution is experimentally justified, in the absence of the possibility of comparing it to an exact solution, thereby suggesting their use as an additional validation metric for the obtained solutions.