This paper retrieves bright 1-soliton solution to the perturbed concatenation model by the application of the semi-inverse variational principle. The perturbation terms are from inter-modal dispersion and self-steepening effect that is considered with maximum allowable intensity. The parameter constraints that naturally emerged from the integration scheme are enumerated.

This paper examines the sixth-order generalized Boussinesq equation, which describes the dynamics of shallow-water waves, including the effects of surface tension. The study introduces Kudryashov’s R-function neural network approach for the first time, aiming to provide exact solutions to the nonlinear differential equation that represents the mathematical model of the sixth-order generalized Boussinesq equation. This technique incorporates the solutions of a nonlinear differential equation into neural networks, using them as an activation function within the hidden layer. In line with previous research on this topic, two categories of solutions are derived: solitary wave and shock wave solutions. Additionally, this paper includes 3D and 2D graphs to visually represent the solutions obtained.

We critically reexamine traveling-wave solutions and Benjamin–Feir instability for a Schrödinger–Hirota equation with power-law nonlinearity. A traveling-wave reduction, Hamiltonian phase-plane analysis, direct substitution, and symbolic residual checks are used to establish the compatibility of candidate solutions with the original complex equation. The previously reported unrestricted extended-Kudryashov solution family is shown to be invalid; only a restricted singular branch survives on a half-line and cannot represent a globally regular optical soliton. A regular bright solitary pulse is verified, and sign-changing dark waves are characterized through an implicit real quadrature rather than ambiguous fractional powers. The corrected Jacobian determinants yield consistent classifications of equilibria and their homoclinic or heteroclinic trajectories. For continuous-wave backgrounds, the perturbation dispersion relation determines the instability band and the corrected maximum growth rate. Explicit parameter sets and reproducibility checks distinguish valid traveling waves from singular profiles and clarify that instability of the background does not establish instability of localized solitons.

This paper provides highly dispersive optical soliton solutions to the perturbed complex Ginzburg–Landau equation. The self-phase modulation structures are maintained in three forms, which are derived from the power law of nonlinearity with arbitrary intensity. The paper employs the semi-inverse variational principle as its integration scheme, as conventional methods are incapable for it. The amplitude–width relation of the solitons is reconstructed by employing Cardano’s method to solve a cubic polynomial equation. Also presented are the necessary parameter constraints that naturally arise from the scheme. These findings enhance our understanding of soliton dynamics and pave the way for further research into more complex nonlinear systems. Future studies may explore the implications of these results in various physical contexts, potentially leading to novel applications in fields such as fiber optics and quantum fluid dynamics.