This research investigates the inverse problem of reconstructing the PT-symmetric potential in a class of (1 + 1)-dimensional nonlinear Schrödinger equations. The governing equation is given by: $\mathrm{i} u_t(x, t)+u_{x x}(x, t)+\alpha|u(x, t)|^2 u(x, t)+\beta|u(x, t)|^4 u(x, t)+V_{P T}(x) u(x, t)=0$ where u(x, t) denotes the wave function in dimensionless coordinates, and the PT-symmetric potential VPT (x) = V (x)+iW(x) consists of a real part V (x) and an imaginary part iW(x), satisfying the symmetry conditions V (x) = V (-x) and W(x) = -W(x)。
In this inverse problem, partial boundary values of the wave function are known, while the potential VPT (x) 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.
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.
We conduct numerical experiments under both noise-free (exact data) and noisy (inexact data) conditions. The results demonstrate that in both cases, the proposed method converges within a limited number of iterations and maintains the reconstruction error within the order of 10-3. These findings validate the effectiveness and robustness of the proposed method in solving inverse problems involving PT-symmetric potentials, offering an innovative and practical approach for related numerical applications.
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