The study of optical solitons has advanced significantly due to their stability and diverse applications, particularly in high-speed telecommunications, optical signal processing, and quantum technologies. This paper focuses on the derivation of exact soliton solutions for the nonlinear Kairat-Ⅱ (K-Ⅱ) equation, which models second-order spatiotemporal and group velocity dispersion effects in nonlinear optical systems. By applying the Kumar–Malik method and the extended hyperbolic function method, a comprehensive set of soliton solutions are obtained, capturing the intricate propagation dynamics of solitons in nonlinear media. The Kumar–Malik method yields numerous soliton solutions such as Jacobi elliptic function solution with an elliptic modulus, the trigonometric function solution, the hyper-trigonometric solution, dark solitons, bright solitons, periodic, singular and rational function solutions, etc. The behavior, stability, and evolution of these solitons are further illustrated through two-dimensional (2D), three-dimensional (3D), and contour plots, providing insights into their structural characteristics under various physical conditions. In this article, new soliton solutions in Jacobi elliptic form are derived for the given equation, providing valuable insights into the theoretical framework of solitons in nonlinear optics and presenting potential advancements for soliton-based technologies.

This work utilized the concepts of fuzzy interval analysis and convexity to explore some novel refinements of classical counterparts. The main goal was to look into a type of strong convexity that connected the ideas of pseudo-ordering, ${\delta }^1$-level mappings, and the control function ${\hslash }_{\circ }$. This type of mapping is called a fuzzy number-valued ${\hslash }_{\circ }$-super-quadratic mapping. An interesting fact is that all the function classes extracted from this class were new and novel and quite useful in the optimization and approximation theory. We assessed this class of functions pertaining to essential properties, examples, and various integral inequalities such as Jensen's, reverse Jensen's, Jensen-Mercer, Hermite-Hadamard and Fejer's like inequalities in the classical, and fractional framework. Furthermore, we delivered the accuracy of our findings through graphical and tabular approaches, particularly a novel application for means.