The optical solitary waves explain non-dispersive and non-diffractive spatiotemporal localized waves envelopes promulgating in the media of optical Kerr. These propagations are generally described by the nonlinear Schro dinger equation. The modified extended mapping technique is utilized to assemble the solitons, solitary waves and rational solutions of time-dependent dimensionless paraxial wave equation. The obtained different sorts of wave solutions encompass key applications in physics and engineering. By giving appropriate values to parameter, special sorts of solitary waves configuration can be displayed graphically. The physical interpretation of the solution can be understood through the structure. The stability of this wave equation is investigated via using modulational instability analysis which authenticates that all soliton solutions are stable and exact. Several analytical results and working out have confirmed the strength and efficacy of the current technique.

中文翻译：

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克尔介质旁轴波动力学模型的椭圆函数解、调制不稳定性和光孤子分析

光学孤立波解释了在光学克尔介质中传播的非色散和非衍射时空局部波包络。这些传播通常由非线性薛定谔方程描述。利用改进的扩展映射技术组装孤子、孤波和瞬态无量纲旁轴波动方程的有理解。获得的不同类型的波解包括物理和工程中的关键应用。通过给参数赋予适当的值，可以以图形方式显示特殊种类的孤立波配置。解的物理解释可以通过结构来理解。该波动方程的稳定性是通过使用调制不稳定性分析来研究的，该分析证明所有孤子解都是稳定和精确的。多项分析结果和计算结果证实了当前技术的强度和功效。