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<p>This paper presents a novel application of an extended tanh approach, termed the NITM (Novel Improved Tanh approach), for tackling the complex cubic nonlinear Schrödinger equation with a delta potential. This model serves as essential for the description of optical waveguides that contain localized defects. Through a unified computational framework, we derive a extensive family of exact analytical solutions, including trigonometric periodic waves, hyperbolic solitons, and solutions expressed in terms of Jacobi and Weierstrass elliptic functions, as well as rational forms. The <Emphasis FontCategory="SansSerif">NITM</Emphasis> framework constitutes a significant extension of classical tanh-function methods by incorporating a more flexible ansatz structure, which proves highly effective in managing complex nonlinearities. All derived solutions are rigorously verified using symbolic computation in Maple. Furthermore, detailed graphical analyses illuminate the influence of critical parameters on wave propagation dynamics. As one of the first implementations of <Emphasis FontCategory="SansSerif">NITM</Emphasis> in the scientific literature, this work not only demonstrates the method’s superior capability for generating a diverse spectrum of solutions with computational efficiency but also provides new insights into nonlinear wave phenomena. These findings confirm NITM as a robust tool for the analysis of nonlinear partial differential equations in mathematical physics.</p>

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A New Exploration of Some Explicit Soliton Solutions to a Modified Nonlinear Schrödinger Equation

  • 期刊论文

  • 发表日期：

    2026年1月6日

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  • Hongzhi Bao,
  • Foroud Parvaneh,
  • Shami A. M. Alsallami

摘要

This paper presents a novel application of an extended tanh approach, termed the NITM (Novel Improved Tanh approach), for tackling the complex cubic nonlinear Schrödinger equation with a delta potential. This model serves as essential for the description of optical waveguides that contain localized defects. Through a unified computational framework, we derive a extensive family of exact analytical solutions, including trigonometric periodic waves, hyperbolic solitons, and solutions expressed in terms of Jacobi and Weierstrass elliptic functions, as well as rational forms. The NITM framework constitutes a significant extension of classical tanh-function methods by incorporating a more flexible ansatz structure, which proves highly effective in managing complex nonlinearities. All derived solutions are rigorously verified using symbolic computation in Maple. Furthermore, detailed graphical analyses illuminate the influence of critical parameters on wave propagation dynamics. As one of the first implementations of NITM in the scientific literature, this work not only demonstrates the method’s superior capability for generating a diverse spectrum of solutions with computational efficiency but also provides new insights into nonlinear wave phenomena. These findings confirm NITM as a robust tool for the analysis of nonlinear partial differential equations in mathematical physics.