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<p>This article presents a hybrid neural network–finite difference method (NN-FDM), developed for the first time to solve a nonlinear Schrödinger equation with Kerr law nonlinearity and weak nonlocality (SEWKLNAWN). Analytical solutions of the SEWKLNAWN were previously derived by Wazwaz. Building on these results, the present study proposes a novel computational framework that combines finite difference decomposition with a neural network approach to obtain accurate numerical solutions of the model. The proposed NN-FDM integrates the stability and physical consistency of classical finite difference schemes (FDS) with the powerful approximation capability of neural networks. Numerical simulations are conducted, and the obtained results are systematically compared with analytical solutions available in the literature. The comparisons demonstrate excellent agreement between the NN-FDM solutions and the exact results, confirming the accuracy, robustness, and efficiency of the proposed method. These findings indicate that the NN-FDM provides a reliable and efficient tool for the numerical investigation of nonlinear Schrödinger-type equations arising in nonlinear optics and related fields.</p>

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A new hybrid neural-network finite difference method for a nonlinear Schrödinger equation with Kerr law nonlinearity and weak nonlocality

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月17日

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  • Mahmut Modanli

摘要

This article presents a hybrid neural network–finite difference method (NN-FDM), developed for the first time to solve a nonlinear Schrödinger equation with Kerr law nonlinearity and weak nonlocality (SEWKLNAWN). Analytical solutions of the SEWKLNAWN were previously derived by Wazwaz. Building on these results, the present study proposes a novel computational framework that combines finite difference decomposition with a neural network approach to obtain accurate numerical solutions of the model. The proposed NN-FDM integrates the stability and physical consistency of classical finite difference schemes (FDS) with the powerful approximation capability of neural networks. Numerical simulations are conducted, and the obtained results are systematically compared with analytical solutions available in the literature. The comparisons demonstrate excellent agreement between the NN-FDM solutions and the exact results, confirming the accuracy, robustness, and efficiency of the proposed method. These findings indicate that the NN-FDM provides a reliable and efficient tool for the numerical investigation of nonlinear Schrödinger-type equations arising in nonlinear optics and related fields.