This article secures new soliton solutions of the higher-order nonlinear Schrödinger equation (NLSE) and the nonlinear Kudryashov's equation by means of two analytical techniques, namely the extended ($\frac{G^{\prime }}{G^2}$)-expansion method and the first integral method. Many exact traveling wave solutions such as hyperbolic function solutions, trigonometric function solutions and rational function solutions with free parameters are characterized. Periodic solitons, dark solitons, singular solitons, combo solitons and plane waves are obtained. The existence criteria for such solutions are also provided. Moreover, the modulation instability analysis is used to examine the stabilities of both equations, by which the modulation instability gain spectrums of both equations are obtained. The obtained solutions are also presented graphically.

本文通过两种分析技术，即扩展的（$\frac{G^{\prime }}{G^2}$）展开法和第一个积分法。刻画了许多精确的行波解，例如双曲函数解，三角函数解和带自由参数的有理函数解。获得了周期孤子，暗孤子，奇异孤子，组合孤子和平面波。还提供了此类解决方案的存在标准。此外，使用调制不稳定性分析来检查两个方程的稳定性，从而获得两个方程的调制不稳定性增益谱。所获得的解决方案也以图形方式呈现。