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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: $ \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_{PT}(x) u(x,t) = 0, $ where $ u(x, t) $ denotes the wave function in dimensionless coordinates, and the PT-symmetric potential $ V_{PT}(x) = V(x) + {\bf{i}}W(x) $ consists of a real part $ V(x) $ and an imaginary part $ {\bf{i}}W(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 $ V_{PT}(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. -
Keywords:
- PT-symmetry /
- Schrödinger equation /
- potential function /
- inverse problem
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图 1 $ V_{pt}(x) $实部及虚部反演结果 图(a) $ V_{pt}(x) $实部的实际精确值与INTA和mPINNs反演结果的对比; 图(b) $ V_{pt}(x) $虚部的实际精确值与INTA和mPINNs反演结果的对比; 图(c) INTA反演结果实部与虚部的绝对误差
Figure 1. Real and imaginary part inversion results of $ V_{pt}(x) $ (a) Comparison of the real part of $ V_{pt}(x) $ with the inversion results of INTA and mPINNs; (b) Comparison of the imaginary part of $ V_{pt}(x) $ with the inversion results of INTA and mPINNs; (c) Absolute error of the real and imaginary parts of INTA inversion results.
图 2 误差 (a) INTA每次迭代后的误差error; (b) mPINNs与精确解$ u(x, z) $之间的误差
Figure 2. Errors (a) error after each iteration of INTA; (b) Error between mPINNs and the exact solution $ u(x, z) $.
图 3 $ V_{pt}(x) $实部及虚部反演结果($ U^* $包含噪声) (a) INTA和mPINNs反演结果与$ V_{pt}(x) $实部的实际精确值的对比; (b) INTA和mPINNs反演结果与$ V_{pt}(x) $虚部的实际精确值的对比; (c) INTA反演结果实部与虚部的绝对误差
Figure 3. Real and imaginary part inversion results of $ V_{pt}(x) $ ($ U^* $ with noise) (a) Comparison of the real part of $ V_{pt}(x) $ with the inversion results of INTA and mPINNs; (b) Comparison of the imaginary part of $ V_{pt}(x) $ with the inversion results of INTA and mPINNs; (c) Absolute error of the real and imaginary parts of INTA inversion results.
图 4 误差($ U^* $包含噪声) (a) INTA每次迭代后的误差error; (b) mPINNs与精确解$ u(x, z) $之间的误差
Figure 4. Errors($ U^* $ with noise) (a) error after each iteration of INTA; (b) Error between mPINNs and the exact solution $ u(x, z) $.
表 1 INTA与mPINNs对比
Table 1. Comparison between INTA and mPINNs.
算法 实部误差 虚部误差 运行时间(秒) INTA $ 6.1768\times10^{-13} $ $ 7.6891\times10^{-12} $ 61.3 mPINNs 0.0416 0.01549 113.8 
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表 2 绝对误差
Table 2. Absolute Error.
x $ 0.1 $ $ 0.2 $ $ 0.3 $ $ 0.4 $ $ 0.5 $ $ 0.6 $ $ 0.7 $ $ 0.8 $ $ 0.9 $ INTA实部 $ 6.4\text{E-13} $ $ 6.5\text{E-13} $ $ 6.1\text{E-13} $ $ 6.0\text{E-13} $ $ 6.1\text{E-13} $ $ 6.4\text{E-13} $ $ 7.3\text{E-13} $ $ 7.8\text{E-13} $ $ 8.0\text{E-13} $ mPINNs实部 $ 5.3\text{E-03} $ $ 8.8\text{E-03} $ $ 8.6\text{E-03} $ $ 4.8\text{E-03} $ $ 1.5\text{E-03} $ $ 8.5\text{E-03} $ $ 1.4\text{E-02} $ $ 1.6\text{E-02} $ $ 1.1\text{E-02} $ INTA虚部 $ 1.4\text{E-12} $ $ 1.6\text{E-12} $ $ 1.3\text{E-12} $ $ 7.5\text{E-13} $ $ 5.2\text{E-13} $ $ 4.7\text{E-13} $ $ 6.8\text{E-13} $ $ 9.5\text{E-13} $ $ 5.5\text{E-13} $ mPINNs虚部 $ 1.4\text{E-02} $ $ 1.7\text{E-03} $ $ 4.0\text{E-03} $ $ 2.3\text{E-03} $ $ 4.0\text{E-03} $ $ 5.3\text{E-03} $ $ 6.1\text{E-03} $ $ 5.7\text{E-03} $ $ 4.3\text{E-03} $
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表 3 INTA与mPINNs对比($ U^* $包含噪声)
Table 3. Comparison between INTA and mPINNs ($ U^* $ with noise).
算法 实部误差 虚部误差 运行时间(秒) INTA 0.0043 0.0126 71.5 mPINNs 0.0098 0.0452 112.3
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表 4 绝对误差($ U^* $包含噪声)
Table 4. Absolute Error($ U^* $ with noise).
x $ 0.1 $ $ 0.2 $ $ 0.3 $ $ 0.4 $ $ 0.5 $ $ 0.6 $ $ 0.7 $ $ 0.8 $ $ 0.9 $ INTA实部 $ 1.9\text{E-04} $ $ 2.3\text{E-03} $ $ 4.2\text{E-03} $ $ 4.4\text{E-03} $ $ 2.8\text{E-03} $ $ 2.8\text{E-03} $ $ 3.1\text{E-03} $ $ 5.2\text{E-03} $ $ 3.2\text{E-03} $ mPINNs实部 $ 8.9\text{E-03} $ $ 1.1\text{E-02} $ $ 1.2\text{E-02} $ $ 1.1\text{E-02} $ $ 8.5\text{E-03} $ $ 5.9\text{E-03} $ $ 3.5\text{E-03} $ $ 1.9\text{E-03} $ $ 1.3\text{E-03} $ INTA虚部 $ 3.5\text{E-03} $ $ 2.3\text{E-03} $ $ 2.6\text{E-03} $ $ 8.7\text{E-04} $ $ 2.3\text{E-04} $ $ 1.8\text{E-03} $ $ 2.4\text{E-03} $ $ 3.8\text{E-03} $ $ 1.8\text{E-03} $ mPINNs虚部 $ 2.4\text{E-03} $ $ 6.2\text{E-03} $ $ 1.0\text{E-02} $ $ 1.6\text{E-02} $ $ 1.6\text{E-02} $ $ 1.6\text{E-02} $ $ 1.5\text{E-02} $ $ 1.2\text{E-02} $ $ 7.1\text{E-03} $
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