Abstract
In this study, the monochromatic beam propagation interprets non-scattering and non-dissipation spatiotemporally localized wave parcels proliferated in the announcement of optical Kerr. By using the mathematical techniques some elliptic, rational, and soliton solutions of dimension-less (time-dependent) paraxial wave structure are established. The Sardar subequation method (SSM) and mathematica 11.0 are used to find the exact solution of the paraxial wave equation. The solutions which we will be obtained explain some key implementations in engineering and physics. Some graphical representation has been explained in modulus, real and imaginary graph by considering relevant values of the framework. The solidity of this work explains that the soliton solutions are secure and perfect.