Numerical methods for solving the inverse problem of 1D and 2D PT-symmetric potentials in the NLSE

Abstract

This paper establishes a numerical framework for addressing the inverse problem of PT-symmetric potentials. Firstly, we discretize the solution space and innovatively construct a mapping to project the inverse problem of the PT-symmetric potential onto a finite-dimensional real vector space, thereby transforming the inverse problem of PT-symmetric potentials in the complex domain into a root-finding problem for a system of nonlinear equations in the real-number domain. Subsequently, to address the ill-posedness of the equation system, we innovatively apply regularization techniques and numerical algebraic techniques, constructing the Regularized-Newton-GMRES method for solving nonlinear equation systems, thereby obtaining the regularized solution for the PT-symmetric potential inverse problem. Finally, we conduct numerical experiments to validate the effectiveness of the established numerical solution framework. Our numerical experiments demonstrate that the proposed Regularized-Newton-GMRES method achieves higher computational accuracy, shorter computation time, improved stability, and effective solutions for such inverse problems.