Adverse impacts of multiplicative noise on the exact solutions of the generalized nonlinear Schrödinger equation derived using two approaches

The purpose of this study is to investigate the effects of multiplicative noise on the exact solutions of the generalized nonlinear Schrödinger equation. By employing two direct methods, we derive precise solutions, including hyperbolic, trigonometric and Jacobi elliptic function solutions. The research aims to enhance understanding of how multiplicative noise influences the behavior of these solutions, providing valuable insights into the dynamics of stochastic systems. Graphical examples illustrate the impact of noise, contributing to the broader field of nonlinear dynamics and its applications in physics and engineering. This study employs a two-pronged approach to derive exact solutions of the generalized nonlinear Schrödinger equation influenced by multiplicative noise. Initially, a similarity transformation reduces the stochastic problem to a second-order cubic nonlinear ordinary differential equation. Subsequently, four mappings between the Riccati equation and the reduced ordinary differential equation are established using the generalized unified method. The solutions generated include hyperbolic, trigonometric and Jacobi elliptic functions. The behavior of these solutions under varying levels of multiplicative noise is analyzed, supported by graphical representations to illustrate the effects of noise on the derived solutions. The findings reveal that multiplicative noise significantly affects the exact solutions of the generalized nonlinear Schrödinger equation. The study successfully derives multiple families of solutions, including hyperbolic, trigonometric and Jacobi elliptic functions, demonstrating the versatility of the employed methods. The analysis shows that the introduction of noise alters the stability and dynamics of these solutions, leading to complex behaviors. Graphical examples illustrate the impact of varying noise levels, highlighting the importance of considering stochastic effects in nonlinear systems. These results contribute to a deeper understanding of noise-induced phenomena in mathematical physics and engineering applications. This study offers original contributions by exploring the effects of multiplicative noise on the exact solutions of the generalized nonlinear Schrödinger equation, an area that has received limited attention in existing literature. By employing innovative methods, including the generalized unified technique and Jacobi elliptic function method, the research generates a diverse set of solutions that enhance the understanding of stochastic dynamics in nonlinear systems. The findings provide valuable insights into the interplay between noise and soliton behavior, which can inform future research and applications in fields such as optics, plasma physics and complex systems.