We present exact bright, dark and rogue soliton solutions of generalized higher-order nonlinear Schrödinger equation, describing the ultrashort beam propagation in tapered waveguide amplifier, via a similarity transformation connected with the constant-coefficient Sasa-Satsuma and Hirota equations. Our exact analysis takes recourse to identify the allowed tapering profile in conjunction with appropriate gain function which corresponds to $\mathcal{PT}$-symmetric waveguide. We extend our analysis to study the effect of tapering profiles and higher-order terms on the evolution of self-similar waves and thus enabling one to control the self-similar wave structure and dynamical behavior.

我们给出了广义高阶非线性Schrödinger方程的精确的亮，暗和无赖孤子解，它们通过与常数系数Sasa-Satsuma和Hirota方程相连的相似变换，描述了锥形波导放大器中超短光束的传播。我们的精确分析将利用适当的增益函数来确定允许的渐缩曲线，该增益函数对应于$\mathcal{PT}$对称波导。我们扩展了我们的分析，以研究渐缩轮廓和高阶项对自相似波的演化的影响，从而使人们能够控制自相似波的结构和动力学行为。