Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi></mml:mi> <mml:mi></mml:mi> </mml:math> -symmetric Scarf-II potential via PINN deep learning

Abstract

Abstract In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic"></mml:mi> <mml:mi mathvariant="italic"></mml:mi> </mml:math> -symmetric Scarf-II potential in the deep learning.

Topics & Concepts