Predicting positon solutions of a family of Nonlinear Schr\"{o}dinger equations through Deep Learning algorithm

TL;DR

This paper employs Physics-Informed Neural Networks to accurately predict positon solutions across various nonlinear Schrödinger equations, including higher-order and coupled systems, and introduces new exact solutions via Darboux transformation.

Contribution

The study extends PINN applications to complex NLSEs and reports novel exact positon solutions for sextic and coupled generalized NLSEs, validated by analytical comparisons.

Findings

PINN accurately predicts positon solutions with low mean squared error.

The approach successfully handles higher-order and coupled NLSEs.

New exact solutions for sextic and coupled NLSEs are constructed and validated.

Abstract

We consider a hierarchy of nonlinear Schr\"{o}dinger equations (NLSEs) and forecast the evolution of positon solutions using a deep learning approach called Physics Informed Neural Networks (PINN). Notably, the PINN algorithm accurately predicts positon solutions not only in the standard NLSE but also in other higher order versions, including cubic, quartic and quintic NLSEs. The PINN approach also effectively handles two coupled NLSEs and two coupled Hirota equations. In addition to the above, we report exact second-order positon solutions of the sextic NLSE and coupled generalized NLSE. These solutions are not available in the existing literature and we construct them through generalized Darboux transformation method. Further, we utilize PINNs to forecast their behaviour as well. To validate PINN's accuracy, we compare the predicted solutions with exact solutions obtained from analytical methods. The results show high fidelity and low mean squared error in the predictions generated by our PINN model.