This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

中文翻译：

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Ablowitz-Kaup-Newell-Segur 水波方程的孤立波解和 Painleve 分析

本研究的重点是 Ablowitz-Kaup-Newell-Segur (AKNS) 水波方程。将实施 Painleve 检验 (P-test) 以检查 AKNS 方程的可积性，并实施扩展的修正辅助方程映射 (EMAEM) 架构以获得一组新的行波解，如周期性和双周期、钟形、扭结、奇异扭结、反扭结、三角、奇异、有理、组合孤子样解和双曲解。此外，分析了所实现的算法对于求解非线性演化方程（NLEEs）是有效和准确的。最后，还提供了图形模拟（2D、3D 和轮廓）来说明解决方案的详细行为和所提出方法的有效性。