This article deals with coupled nonlinear stochastic partial differential equations. It is a reaction-diffusion system, known as the stochastic Gray-Scott model. The numerical approximation of the stochastic Gray-Scott model is discussed with the proposed stochastic forward Euler (SFE) scheme and the proposed stochastic non-standard finite difference (NSFD) scheme. Both schemes are consistent with the given system of equations. The linear stability analysis is discussed. The proposed SFE scheme is conditionally stable and the proposed stochastic NSFD is unconditionally stable. The convergence of the schemes is also discussed in the mean square sense. The simulations of the numerical solution have been obtained by using the MATLAB package for the various values of the parameters. The effects of randomness are discussed. Regarding the graphical behavior of the stochastic Gray-Scott model, self-replicating behavior is observed.

In this study, the improved tan $\left(\frac{\mathrm{\Omega }(\mathrm{{\rm Y}})}{2}\right)$-expansion method is used to construct a variety of precise soliton and other solitary wave solutions of the Gardner equation. Gardner equation is extensively utilized in plasma physics, quantum field theory, solid-state physics and fluid dynamics. It is the simplest model for the description of water waves with dual power law nonlinearity. Hyperbolic, exponential, rational and trigonometric traveling wave solutions are obtained. The retrieved solutions include kink solitons, bright solitons, dark-bright solitons and periodic wave solutions. The efficacy of this method is determined by the comparison of the newly obtained results with already reported results.

We present a new concept of fractional curvature invariant for regular curves in the Lorentz plane by generalizing the Caputo‐fractional curvature from Euclidean geometry to the pseudo‐Riemannian setting. Our construction projects the integer-order derivative of the Caputo vector of fractional-order derivatives onto the Lorentzian normal direction, yielding a curvature measure that naturally distinguishes timelike and spacelike curves. Explicit formulas for representative model curves are derived, and we illustrate how the Lorentzian metric signature fundamentally changes fractional curvature behavior. This framework extends fractional‐order geometric analysis into relativity, providing new tools for studying memory effects and nonlocal dynamics along curves in relativistic contexts.

This paper discusses an elaborate investigation of the dimensionless time-dependent paraxial equation based on its varied soliton solutions obtained through an improved modified extended tanh function method. The approach is capable of producing kink, bell-shaped, singular wave, periodic, singular periodic wave, bright and dark solitary wave, breather soliton, singular bell, M-shaped, W-shaped, and V-pattern solitons expressed as hyperbolic and trigonometric functions. Visualization via three dimensional (3D) surface plots, two dimensional (2D) cross-sections and contour plots allows a thorough examination of the wave morphology. Nonlinear dynamics are investigated using bifurcation, chaotic, sensitivity, and stability analyses that uncover complex solution behaviors and stability regimes. Modulation instability analysis verifies the stability of soliton structures under perturbations. The work demonstrates the effectiveness of the improved modified extended tanh function method for solving nonlinear partial differential equations and enhances theoretical knowledge of paraxial wave phenomena.

Many years of research have gone into spline functions, and they are now used in countless computational tasks. Splines have a lot of useful properties that make them an excellent tool for numerical problem solving, which account for their never-ending applications. The piecewise continuous functions known as spline functions yield smooth outcomes. The numerical solution to the nonhomogeneous time-fractional Banjamin-Bona-Mahony-Burger problem was presented in this study. The objective of the study was to obtain accurate numerical results by applying the Atangana-Baleanu fractional derivative with the help of the forward difference scheme for integer-order time derivative while the $\theta$-weighted scheme with the collaboration of cubic B-spline functions was used for the spatial derivatives. The stability of the proposed scheme was analyzed and proved to be unconditionally stable. The convergence analysis was also studied, and it was of the second order $O(h^2+(\mathrm{\Delta }s{)}^2)$. The proposed scheme was applicable and accurate, as demonstrated by numerical examples and their conceivable outcomes. The proposed scheme provided accuracy compared to other numerical techniques because it yielded numerical solutions in $C^2$ continuous piecewise form at each knot in the domain.

This work used the modified extended direct algebraic expansion method to find exact soliton solutions for the (2+1)-dimensional nonlinear Zoomeron equation. The modified extended direct algebraic technique employs a wave transformation and, in order to determine solutions, it then performs an algebraic expansion, compares coefficients, and balances the equation. The results were an effective acquisition of a variety of solitons with unique wave characteristics including bright, kink, periodic, singular periodic, and dark solitons. A stability investigation has confirmed the structural integrity of these solutions under minor perturbations. In the form of 2D, contour, and 3D graphical representations, the stability and propagation of these solutions were further investigated. The findings illustrate how effectively this technique can solve higher-dimensional nonlinear equations and yield more soliton solutions. Beyond broadening our knowledge of nonlinear wave behavior, this research could be beneficial in nonlinear optics, fluid motion, and plasma systems.