Exploring stochastic solitons: Variational approaches, conservation laws, and bifurcation of the (3 + 1)-dimensional nonlinear Schrödinger equation with fractional-order numerical insights.

In this paper, we investigate the stochastic (3 + 1)-dimensional nonlinear Schrödinger equation (NLSE), a critical model for understanding wave dynamics under the influence of random perturbations. We apply He’s semi-inverse method to establish a variational formula for the stochastic (3 + 1)-dimensional NLSE and we use this formula to construct conservation laws corresponding to the symmetries of the model. Based on this formulation, a solitary solution can be easily obtained using the Ritz method. We explore approximate solutions of the stochastic (3 + 1)-dimensional NLSE equation by choice of a trial function in the region of the rectangular box in three cases. Subsequently, we approximate the trial function using a piecewise linear ansatz in one case of the two-box potential, followed by an approximation using quadratic polynomials with two free parameters instead of the piecewise linear ansatz. We demonstrate the effectiveness of the analytical approaches, namely, the improved modified extended tanh-function method and He’s semi-inverse variational principle method for seeking more exact solutions via the stochastic (3 + 1)-dimensional NLSE equation. As a result, bright solitons, singular solitons, periodic wave solution and dark solitons solutions are obtained. Furthermore, the dynamical behaviors of the model are discussed by investigating bifurcations at equilibrium points. The paper not only advances the theoretical understanding of the stochastic (3 + 1)D NLSE but also provides practical methodologies for solving this complex equation, offering new insights into wave phenomena influenced by stochastic effects. In addition, we employ the fractional reduced differential transform (F-RDT) method, interpreted through Caputo fractional derivatives, to construct a numerical solution utilizing hyperbolic functions. We further analyze the influence of the fractional order parameter, denoted as α, on the behavior of the numerical solution. This investigation includes a detailed examination of the discrepancies between the exact solitary wave solutions and the corresponding numerical approximations, allowing us to identify the absolute errors. By comparing the two, we provide insights into the accuracy and stability of the numerical approach for solving the studied equation, emphasizing how variations in α impact the precision of the model. [ABSTRACT FROM AUTHOR]