In this research work, the nonlinear coupled Konno–Oono (CKO) system is under examination to explore the dynamical soliton wave structure in the magnetic field through two distinct approaches with qualitative analysis. The magnetic potential represents an essential tool in constructing diverse functions in the fluency of the magnetic field. Based on the nature and physical context of certain magnetic fields, the CKO nonlinear evolution system was raised to modify the magnetic effects. Such impacts on the dynamics for the CKO system depend on the magnetic potential. The current study aims to expand the use of the sine–cosine and tan–cotan methods to process the CKO equations analytically. The two techniques show the applicability, simplicity, and low complexity based on the traveling-wave procedure. Assorted bright, dark, and mixed closed-form soliton solutions have been derived. The 3D solitons with different structures have been depicted. The interaction of displacements inside the magnetic field is also shown. For more generality, and because of the new findings in real-life applications of fractional calculus, the time-fractional CKO model in different versions is discussed, and its effects are illustrated. Finally, we analyze the stability explorations for the corresponding nonlinear ordinary system out of its sensitivity, bifurcation, and chaotic behaviors.

This article investigates certain fixed-point results enjoying nonlinear almost contraction conditions in the setup of relational metric space. Some examples are constructed in order to indicate the profitability of our results. As a practical use of our findings, we demonstrate the existence of a unique solution to a specific first-order boundary value problem.

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation.

The purpose of this study was to investigate the oscillation criteria for nonlinear second-order neutral differential equations with deviating arguments, with a particular emphasis on their non-canonical forms. The primary goal was to expand the current theoretical framework by introducing new relations that improved the monotonicity of positive solutions. To attain this purpose, an iterative technique was used to deduce new oscillation criteria, which helped to enhance present understanding in this field. The study process was based on a thorough review of previous literature, followed by the creation of new oscillation criteria with both theoretical and applied significance. The obtained results were validated by three illustrative instances, demonstrating the importance and influence of these criteria in the study of neutral differential equations, particularly in the study of neutral differential equations, especially in nonlinear contexts.

This paper presents a method for calculating eigenvalues lying in the gaps of the essential spectrum of matrix-valued Schrödinger operators. The technique of dissipative perturbation allows eigenvalues of interest to move up the real axis in order to achieve approximations free from spectral pollution. Some results of the behaviour of the corresponding eigenvalues are obtained. The effectiveness of this procedure is illustrated by several numerical examples.