The current paper retrieves optical 1–soliton solution to the concatenation model with the Kerr law of self-phase modulation structure. The primary mode of integration is the Lie symmetry analysis. The subsequent ordinary differential equations (ODEs) are formulated by the recently proposed Kumar–Malik scheme. This leads to the emergence of a variety of 1–soliton solutions along with their respective existence criteria. The numerical simulations support the analytical solutions that are yielded by the joint application of the two schemes.

We investigate a stochastic nonlinear Schrödinger equation modeling optical bullets in dispersive nonlinear media. By applying Lie-symmetry analysis, the governing system is reduced to ordinary differential equations, enabling exact traveling-wave solutions through the Kumar–Malik and Kudryashov approaches. Phase-plane analysis and bifurcation structures are derived, and the Melnikov method is employed to detect homoclinic chaos under stochastic perturbations. The results demonstrate the coexistence of stable solitary structures and chaotic dynamics, offering new insights into energy localization and control in nonlinear optical systems. The methodology unifies symmetry-based reductions, exact solutions, and nonlinear bifurcation analysis, contributing to the broader theory of stochastic nonlinear evolution equations.

In this study, we investigate systems known as nonlinear fractional delay differential (nLFDD) systems, characterized by finite state delays and fractional orders within the range of 0 < η ≤ 1 < ζ ≤ 2 , situated infinite-dimensional settings. We utilize the controllability Gramian matrix to establish both necessary and sufficient conditions for the controllability of linear fractional delay differential systems that fall within the order range of 0 < η ≤ 1 < ζ ≤ 2 . Moreover, the Schauder fixed point theorem is employed to delineate the sufficient conditions required for the controllability of nLFDD systems, which are defined by finite state delays and fractional orders in the specified range. To substantiate the theoretical constructs put forth, we provide two illustrative examples.