Table 1 Summary of parameters in the SGNLSE: physical meanings and roles.

\(\digamma (x,t)\)

is the complex wave function representing the wave soliton. It evolves under the combined effects of dispersion, nonlinearity, and stochastic perturbations.

\(|\digamma |^{2m}\), \(|\digamma |^{2n}\) and \(|\digamma |^{2p}\)

represent nonlinear interactions between the wave function and its own intensity, and model intensity-dependent dispersion and refractive index behavior.

a

is the nonlinearity coefficient associated with the standard nonlinear term \(a|\digamma |^{2n}\digamma\). It governs self-phase modulation (SPM). A positive a induces focusing nonlinearity, while a negative a induces defocusing effects. Plays a central role in shaping soliton solutions.

b

is the coefficient of the higher-order nonlinear term \(b|\digamma |^{2p}\digamma\), used to model complex nonlinear behaviors such as multi-photon absorption or saturation effects. Influences the richness and diversity of exact solution structures.

\(\delta\)

is the noise strength parameter for the multiplicative stochastic term \(\delta \digamma \, \frac{dW(t)}{dt}\). It determines the intensity of the stochastic influence on the wave, introducing amplitude fluctuations and phase jitter. Higher values lead to greater randomness in soliton behavior.

W(t)

is the Wiener process (Brownian motion), a stochastic process with independent Gaussian increments. Models environmental randomness such as thermal fluctuations or refractive-index disorder. Its derivative formally appears as white noise in the SGNLSE.

m

is power index of the nonlinear dispersion term \((|\digamma |^{2m} \digamma )_{xx}\). It generalizes the dispersive behavior, allowing the equation to model intensity-dependent dispersion observed in certain optical media or plasmas.

n

is power index in the standard nonlinear term \(a|\digamma |^{2n}\digamma\). It generalizes the Kerr nonlinearity and determines the self-focusing/defocusing strength.

p

is power index in the higher-order nonlinear term \(b|\digamma |^{2p}\digamma\). Controls the degree and type of higher-order nonlinearity contributing to the wave dynamics.

\(\beta\)

is spatial scaling parameter introduced in the transformation \(\zeta = \beta x - vt\), where \(\zeta\) is a traveling coordinate. It controls the width and spatial frequency of the soliton and enables conversion of the SGNLSE into a solvable ordinary differential equation.

\(\omega\)

is the temporal frequency term appearing in the solution’s phase factor \(e^{i[\omega t + \delta W(t) - \delta ^2 t]}\). It governs the oscillatory time evolution of the soliton and couples with the noise to influence stability.