On the soliton solutions of the generalized stochastic nonlinear Schrödinger equation with Kerr effect and higher order nonlinearity via two analytical methods

In this study, we investigate the generalized stochastic nonlinear Schrödinger equation, which models the propagation of ultrashort optical pulses in nonlinear and dispersive media, incorporating both Kerr effect and higher-order nonlinearities. To construct exact analytical solutions, we employ the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>tan</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mi>ϖ</m:mi> <m:mo>(</m:mo> <m:mi>ξ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> tan ( {{{ (ξ )} 2}} ) -expansion method and the ( G ′/ G , 1/ G )-expansion method. These methods yield a variety of exact solutions, including dark, singular, and singular periodic soliton solutions, each representing different physical wave behaviors. We further perform a stability analysis to determine the robustness of these solutions under perturbations and examine their temporal evolution to better understand their propagation dynamics. Graphical illustrations of selected solutions are provided to visualize their dynamics and to demonstrate how the passage of time influences the structure and stability of the resulting wave forms.