This article investigates the practically fixed-time synchronization problem for uncertain chaotic systems. First, a practically fixed-time stability criterion is introduced. Based on this criterion, an adaptive backstepping control technique is employed to design the controller. Unlike traditional dynamic surface control strategies, the proposed approach incorporates a continuous piecewise function. Theoretical analysis demonstrates that the proposed control strategy ensures the practically fixed-time stability of the closed-loop system, with the synchronization error converging to a small neighborhood of the origin within a fixed time. Finally, a simulation example is provided to verify the feasibility and effectiveness of the proposed method.

The vibration absorption performance of two kinds of two-degree-of-freedom (TDOF) nonlinear energy sinks (NESs) is studied and compared to that of a single-degree-of-freedom (SDOF) NES. The two TDOF NESs have different stiffness characteristics in their second degree of freedom. One has a purely cubic, and the other one has a bi-stable stiffness characteristic. Complexification-averaging and Runge–Kutta methods are used to investigate the performance of the NESs. The results show that the vibration absorption performance of both TDOF NESs is significantly higher than that of the SDOF NES after the excitation amplitude increases to a certain value, under both harmonic and shock excitations. The performance of all three NESs is similar below this critical value. In addition, the unstable response induced by the SDOF NES is found to precede higher response branches. However, the additional oscillators in the two TDOF NESs cause the unstable responses to revert to stable responses, and the unstable responses reappear and further transformed into higher branches until the excitation amplitude reaches a relatively high value. Moreover, the results demonstrate that the two TDOF NESs have advantages over each other in different excitation ranges. The NES with a bi-stable component is superior to the NES with only purely cubic components in a relatively wide excitation range of harmonic and shock excitations.

The integrable Kuralay equations (IKEs) represent a new class of nonlinear partial differential equations (NLPDEs) that capture the delicate balance between dispersion and nonlinearity in complex physical systems such as nonlinear optical fibers, plasmas, and ferromagnetic spin chains. From a geometric standpoint, these equations correspond to the motion of space curves, where soliton solutions describe the curvature and torsion of evolving wave structures. In this work, we employ the Complete Discrimination System for Polynomial Method (CDSPM) to systematically construct and classify a rich family of optical soliton solutions, including Jacobian elliptic, double-hyperbolic (DH), double-periodic (DP), and trigonometric forms. By exploring the limiting behaviors of elliptic solutions, we also derive solitary wave (SW) structures that exhibit kink-type, periodic, and localized pulse geometries. Furthermore, the bifurcation and stability analyses reveal intricate quasi-periodic and transitional behaviors, highlighting the strong coupling between the system’s geometry and its nonlinear dynamics. The findings not only deepen the theoretical understanding of the Kuralay framework but also offer practical applications in optical communication, plasma energy transport, and nonlinear circuit design. Overall, the study presents a unified analytical and geometrical framework that bridges mathematical theory with real-world physical phenomena.

In this paper, we consider a singular limit of the Chern–Simons–Higgs equations that reveals a correspondence between the Chern–Simons–Higgs and Chern–Simons–Schrödinger systems. Specifically, we demonstrate that in the limit as the speed of light goes infinity, known as the non-relativistic limit, a family of time-dependent solutions of the modulated Chern–Simons–Higgs system, which depends on the parameter c representing the speed of light, converges to the solution of the Chern–Simons–Schrödinger system, given suitable initial data assumptions.

In this paper, we use the Nash-Moser iteration method to prove the local and global gradient estimates of positive solutions to the weighted p-Laplacian equation $$\begin{aligned} \Delta _{p,f} u+au^\sigma =0 \end{aligned}$$defined on a complete smooth metric measure space under the condition that the m-Bakry-Émery Ricci curvature has a lower bound, where $$p>1$$, a and $$\sigma \in \mathbb {R}$$ are constants and $$\Delta _{p,f} u=e^{f}\textrm{div}(e^{-f}|\nabla u|^{p-2}\nabla u)$$ is the weighted p-Laplacian operator. As applications, we derive Liouville type theorems and Harnack inequalities for positive solutions to the above equation.

In this paper, we consider non-isospectral problems of two distinct dimensions on the loop algebra of the symplectic Lie algebra $$\mathfrak {sp}(6)$$, and construct two integrable systems. Furthermore, we derive their Hamiltonian structures using the Tu scheme. Additionally, we construct an integrable hierarchy on the generalized Lie algebra $$\mathfrak {Gsp}(6)$$ and establish its Hamiltonian structure as well.

We study a quasistatic contact problem involving a thermo-viscoelastic rod and a rigid obstacle. The material response follows a nonlinear Kelvin-Voigt law with thermal coupling, where the elasticity operator is locally Lipschitz. A generalized damped condition describes the contact interaction. By formulating the problem variationally, we establish existence and uniqueness of the weak solution through nonlinear evolutionary PDE techniques and fixed-point theory. Finally, we detail our numerical approach and demonstrate its performance through several one-dimensional numerical simulations based on the DGM-FEM algorithm.

This study introduces an efficient numerical scheme for solving the fractional-order FitzHugh-Nagumo (FN) equation under given initial and boundary conditions. Unlike traditional methods that often pair high-order spatial discretization with low-order temporal schemes, our approach ensures a balanced, high-order treatment of both time and space. The method employs fractional Lagrange basis functions, constructed on Chebyshev-Gauss-Lobatto collocation points, to interpolate the solution. This formulation transforms the original fractional differential problem into a nonlinear algebraic system via an operator matrix, which is then solved using the Levenberg-Marquardt algorithm. A rigorous convergence analysis is provided. The method’s superior accuracy and efficiency are demonstrated through six illustrative examples, with favorable comparisons made against several recent numerical techniques. The numerical simulations also yield biological insights: we examine how the neuronal excitation threshold influences the action potential, and we model epileptic neuronal behavior by incorporating a physiologically relevant input function.

The (3+1)-dimensional nonlinear extended Zakharov-Kuznetsov (3-dEZK) equation is a fundamental model in plasma physics and nonlinear wave dynamics, playing a key role in describing shallow water waves, ion-acoustic waves in magnetized plasma, and related physical systems. This dynamic model is also a powerful tool in ocean engineering for modeling complex wave interactions, predicting hazardous wave events, and enhancing coastal and offshore infrastructure design. In this work, we employ two analytical techniques, namely the new modified simple equation (nMSE) method and the generalized Arnous (GA) method, to construct soliton and other wave solutions, including bright and dark interactional waves and multi-peak solitons, for the 3-dEZK equation. These results are significant for the physical interpretation of this dynamic model and have applications in fluid dynamics, optical waveguides, plasma physics, etc. We also use bifurcation analysis to reveal state transitions under different parameter settings, identifying critical thresholds where wave propagation changes qualitatively. Furthermore, sensitivity and chaotic analyses are performed to examine the impact of perturbation terms. The novelty lies in deriving new soliton solutions of the 3-dEZK equation via nMSE and GA methods, along with extended bifurcation, sensitivity, and chaos analyses. The graphical analytical solutions highlight the approach’s effectiveness and practical relevance in applied sciences and engineering.