Numerical inversion of PT-symmetric potential functions for (1+1)-dimensional nonlinear Schrödinger equations

<sec>The inverse problem of reconstructing the PT-symmetric potential in a class of (1+1)-dimensional nonlinear Schrödinger equation is investigated in this work. The governing equation is given as follows:</sec><sec> <inline-formula><tex-math id="Z-20250604162436">\begin{document}$ \text{i}u_t(x,t) + u_{xx}(x,t) + \alpha\left| u(x,t) \right|^2 u(x,t) + \beta\left| u(x,t) \right|^4 u(x,t) + V_{\rm PT}(x) u(x,t) = 0, $\end{document}</tex-math></inline-formula>where <i>u</i>(<i>x</i>, <i>t</i>) denotes the wave function in dimensionless coordinates, and the PT-symmetric potential <i>V</i><sub>PT</sub>(<i>x</i>) = <i>V</i>(<i>x</i>)+i<i>W</i>(<i>x</i>) consists of a real part <i>V</i>(<i>x</i>) and an imaginary part i<i>W</i>(<i>x</i>), satisfying the symmetry conditions <i>V</i>(<i>x</i>) = <i>V</i>(–<i>x</i>) and <i>W</i>(<i>x</i>) = –<i>W</i>(<i>x</i>).</sec><sec>In this inverse problem, partial boundary values of the wave function are known, while the potential <inline-formula><tex-math id="M8">\begin{document}$ V_{\rm PT}(x) $\end{document}</tex-math></inline-formula> is the unknown to be reconstructed. To address this challenge, we construct a three-level finite difference scheme for the corresponding forward problem, discretizing both the wave function and the potential. This approach leads to a nonlinear system of equations that links the known wave data to the unknown potential values. To simplify the computation, we separate the real and imaginary parts of this system and reformulate it as a real-valued nonlinear system of equations.</sec><sec>For the numerical solution, we employ an inexact Newton method to iteratively solve the resulting nonlinear system. In each iteration, the Jacobian matrix is approximated numerically. To ensure that the reconstructed potential strictly satisfies the PT-symmetry, a parity correction mechanism is introduced at the end of the iteration process.</sec><sec>We conduct numerical experiments under both noise-free (exact data) and noisy (inexact data) conditions. The results indicate that in both cases, the proposed method converges within a limited number of iterations and keeps the reconstruction error within the order of 10<sup>–3</sup>. These findings verify the effectiveness and robustness of this method in solving inverse problems of PT-symmetric potentials, and provide a new idea and powerful method for related numerical applications.</sec>