Soliton,breather,and rogue wave solutions for solving the nonlinear Schr(o)dinger equation using a deep learning method with physical constraints

• 作者: Jun-Cai Pu Jun Li Yong Chen
• 来源期刊: 中国物理B（英文版）
• 年，卷（期）: 2021,（6）
• 所属期刊栏目: GENERAL
• 页码范围: 86-97
• 页数: 12页
• 语种: 英文
• DOI: 10.1088/1674-1056/abd7e3

摘要

The nonlinear Schr(o)dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimen-sions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation's dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schr(o)dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of non-linear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soli-ton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schr(o)dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.