In this paper, the novel exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved $cosh(\mathrm{\Gamma }(\varpi ))-\mathrm{\text{sech}}(\mathrm{\Gamma }(\varpi ))$ function scheme and the combined $cos(\mathrm{\Gamma }(\varpi ))-sec(\mathrm{\Gamma }(\varpi ))$ function scheme are found. Diverse collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. Received solutions are recast in several hyperbolic, rational and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. We demonstrated that these solutions validated the program using Maple and found them correct. The proposed methodology for solving the metamaterials model has been designed to be effectual, unpretentious, expedient and manageable. Applications of the solutions by the mentioned techniques will be useful to investigate the signals properties of optical fibers, plasma physics phenomena, electromagnetic fields occurrences and various types of nonlinear metamaterials models.

在本文中，具有抛物线非线性 (NL) 定律的广义非线性薛定谔方程的新颖精确孤波解采用改进的$科什(\mathrm{\Gamma }(\varpi ))-\mathrm{\text{秒}}(\mathrm{\Gamma }(\varpi ))$功能方案和组合$因(\mathrm{\Gamma }(\varpi ))-秒(\mathrm{\Gamma }(\varpi ))$找到功能方案。获得的各种双曲和三角函数解的集合依赖于所考虑的方程和辅助 ODE 之间的映射。根据方程中涉及的参数和解中出现的积分常数之间的不同限制，将接收到的解重新转换为几种双曲、有理和三角形式。报告的解决方案中的一些重要解决方案被描绘成利用数学软件感知所考虑模型的物理效用和特殊性。这项工作的主要主题是，人们可以可视化和更新知识以克服最常见的技术并击败解决 ODE 和 PDE。我们证明了这些解决方案使用 Maple 验证了该程序并发现它们是正确的。所提出的解决超材料模型的方法被设计为有效、朴素、方便且易于管理。通过上述技术应用解决方案将有助于研究光纤的信号特性、等离子体物理现象、电磁场发生和各种类型的非线性超材料模型。