In this manuscript, a (3+1)-dimensional nonlinear evolution equation (EE), which is a generalization of the (3+1)-dimensional Hirota bilinear equation and the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation, is explored. The Hirota direct method, along with the Cole-Hopf transformation, is utilized to achieve the bilinearization of the considered equation. Via a suitable auxiliary function (AF) and bilinear form, multiple soliton and lump solutions and their interaction with kink solitons are derived. Moreover, the chaotic behavior of the acquired soliton solutions is demonstrated in the light of the Duffing chaotic system. All the results are portrayed using 3D and density plots to show the behavior of the obtained solutions.

This study develops and analyzes a four-state nonlinear policy–feedback dynamical system that couples a system stressor, an accumulated burden, a signed mitigation–response variable, and a signed policy-pressure variable. The proposed model represents governance response through a smooth threshold-centered feedback mechanism, in which the policy-pressure dynamics depend continuously on the deviation of the stressor from a prescribed reference threshold. Unlike reduced-order formulations with purely exogenous interventions, the present framework generates endogenous interactions among stress accumulation, burden evolution, mitigation response, and policy adjustment. The qualitative analysis establishes local well-posedness in the admissible phase domain, conditional nonnegativity of the accumulated burden, and boundedness of trajectories on admissible intervals. An autonomous effective system is then derived to characterize quasi-stationary mean behavior of the periodically forced dynamics. For this effective system, local stability is investigated using Gershgorin estimates and Routh–Hurwitz criteria, leading to explicit analytical conditions for local asymptotic stability and a critical policy-responsiveness threshold associated with possible Hopf-type oscillatory transitions. The analysis highlights the stabilizing role of mitigation damping and cubic saturation in regulating the feedback loop. To approximate the nonlinear system, a Physics-Informed Neural Network (PINN) surrogate is constructed by embedding the governing equations into a differentiable residual loss while enforcing the initial conditions analytically. The accumulated burden is represented through an admissible neural-network ansatz to preserve the well-definedness of the logarithmic coupling term, while the mitigation–response and policy-pressure variables remain signed in accordance with the model formulation. Numerical validation against reference ode45 solutions across two governance regimes shows maximum absolute errors of order ${10}^{-3}$, indicating that the PINN provides a reliable differentiable surrogate for the coupled policy–feedback dynamics. The resulting framework offers a foundation for future inverse modeling, parameter estimation, and data-assimilation studies involving policy responsiveness, intervention thresholds, and burden- suppression effects.

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables.

This paper investigates refined Hardy-type inequalities on time scales associated with nonnegative kernels and superquadratic functions. Starting from a generalized Hardy operator, we derive an inequality involving explicit remainder terms through a combination of superquadratic Jensen-type estimates and an algebraic power inequality. The resulting estimate is formulated within a time-scale measure framework and is subsequently specialized to power and exponential-type functions. Several consequences, including Hardy–Hilbert- and Pólya–Knopp-type forms, are obtained under appropriate choices of the kernel and the weight. The relation between the proposed inequality and selected results from the existing literature is examined through continuous and time-scale specializations.

This paper establishes several new bivariate Hardy-type dynamic integral inequalities on arbitrary time scales. By utilizing delta integration, properties of rd-continuous functions, and kernel estimates, we provide a unified framework that extends existing continuous two-dimensional inequalities and generalizes their one-dimensional dynamic counterparts. The established inequalities successfully bridge the gap between continuous and discrete analysis, reducing to classical continuous cases when $\mathbb{T}=\mathbb{R}$ and to their discrete analogues when $\mathbb{T}=\mathbb{N}$. These results offer powerful analytical tools for establishing qualitative properties of multi-dimensional dynamic equations.

In this paper, we integrate the concepts of spiral-like functions and Janowski-type functions within the framework of q-calculus to define new subclasses in the open unit disk. Our primary focus is on analyzing convolution conditions that form the foundation for further theoretical developments. The main contributions include establishing sufficient conditions and applying Robertson’s theorem. Furthermore, motivated by a recent definition, we propose analogous neighborhood concepts for the above-mentioned classes, and we explore the corresponding neighborhood results.