A novel method for solving third-order nonlinear Schrödinger equation by deep learning

Abstract

The nonlinear Schrödinger equation is a classic integrable equation, especially plays an important role in nonlinear optics. Therefore, a large number of scholars have studied the solution of this equation and proposed many effective solutions, such as Hirota Bilinear transformation method, Darboux transformation, neural network, and so on. This paper obtain numerical solutions by improving physics-informed neural network (IPINN) method which embeds the physical law, an adaptive activation function and slope recovery term into a traditional neural network. This method can simulate the soliton solution and the rogue wave solution of the nonlinear Schrödinger equation with very little data. The Adam and L-BFGS optimizer are used to optimize network parameters to minimize the loss function. The dynamic behavior of soliton and rogue wave solution of the third-order nonlinear Schrödinger equation is first revealed by using the IPINN method, and the errors obtained by this method are compared with the PINN method. Numerical experiments indicate that this method has better numerical results than the PINN method, which opens up a new way to simulate other physical problems by using the IPINN method.

Keywords:

• Neural network
• Schrödinger equation
• Soliton solution
• Rogue wave solution

Availability of data and code

The datasets generated or analyzed during the current study are available from the corresponding author on reasonable request.

Disclosure statement

The authors declare that they have no conflict of interest.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (11571008, 12061054). Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-20-A06).