In this study, the auxiliary equation method is applied successfully to the Lonngren wave equation. Bright soliton, bright–dark soliton solutions are produced, which play an important role in the distribution and distribution of electric charge. In the conclusion and discussion section, the effect of nonlinearity term on wave behavior in bright soliton traveling wave solution is examined. The advantages and disadvantages of the method are discussed. While graphs representing the stationary wave are obtained, special values are given to the constants in the solutions. These graphs are presented as 3D, 2D and contour.

中文翻译：

---

由传输线模型产生的朗格伦波动方程明暗孤子解的模拟

在这项研究中，辅助方程方法成功地应用于Lonngren波动方程。产生明孤子、明暗孤子溶液，它们对电荷的分布和分布起重要作用。在结论和讨论部分，研究了非线性项对亮孤子行波解中波行为的影响。讨论了该方法的优点和缺点。虽然获得了代表驻波的图形，但对解中的常数给出了特殊值。这些图形以 3D、2D 和等高线的形式呈现。