The fractal complex Hirota-dynamical model (CHDM) is proposed and explored qualitatively and quantitatively in this work. The fractal two-scale transformation, combined with the travelling wave transformation and semi-inverse method, is used to extract the variational principle (VP). Upon the VP, the Hamiltonian is obtained. Using the theory of the planar dynamical system (PDS), the phase portraits are depicted and the existence of the soliton solutions of different shapes is probed by applying the bifurcation theory. Besides, the system’s quasi-periodic and chaotic behaviors are also discussed through imposing different external disturbances. In the end, two approximate analytical methods, namely the Hamiltonian-based method (HBM), which originates from the energy conservation, and the variational method (VM) that is based on the VP and Ritz method, are employed to investigate the different soliton wave solutions. Aided by these two methods, the bell-shaped soliton, [Formula: see text]-shape (double bell-shaped) soliton and the periodic wave solutions are found. As expected, some solutions, including the periodic wave and bell shape soliton solutions, correspond with the discussion on the existence of the solutions with different waveform structures. The outlines of the attained solutions are described graphically. Besides, the effect of the fractal order on their behaviors is also elaborated. The ideas of this work can be employed to explore the solutions of other fractal NPDEs.

This work investigates the existence and uniqueness of a solution to a discrete Robin boundary value problem involving the anisotropic \(\vec{p}\)-mean curvature operator. The existence result is established through variational methods, specifically by applying the Mountain Pass Theorem of Ambrosetti and Rabinowitz in combination with Ekeland’s Variational Principle. Uniqueness is obtained under the assumption of Lipschitz continuity on the nonlinear term.

Theories of emergent gravity have established a deep connection between entropy and the geometry of spacetime by looking at the latter through a thermodynamic lens. In this framework, the macroscopic properties of gravity arise in a statistical way from an effective small-scale discrete structure of spacetime and its information content. In this review, we begin by outlining how theories of quantum gravity imply the existence of a minimum length of spacetime as a general feature. We then describe how such a structure can be implemented in a way that is independent from the details of the quantum fluctuations of spacetime via a bi-tensorial quantum metric qαβ(x,x′) that yields a finite geodesic distance in the coincidence limit x→x′. Finally, we discuss how the entropy encoded by these microscopic degrees of freedom can give rise to the field equations for gravity through a thermodynamic variational principle.

The zig-zag optical lattice model, which plays a major role in quantum physics, is considered in this research. With the aid of the semi-inverse method (SIM), the variational principle (VP) is developed via introducing the trial-Lagrange function (TLF). The detailed derivation process is given and the correctness of the established VP is verified with the Euler–Lagrange equations that were found via computing the stationary conditions. As far as the authors know, the VP is first probed and explored via the SIM in this study. The VP can unfold the intrinsic connections of the different physical quantities, help us better understand the essence of physical phenomena and provide new insights into the exploration and application of the variational approach.

The compaction simulation of compressible delay interbed is an important part of land subsidence simulation. Currently, the most widely used MODFLOW software has two modules, SUB and CSUB, both of which can simulate compressible delay interbed. The difference lies in that the head diffusion equation of the SUB module is based on the principle of head change, while CSUB can use either head change or geological stress variation principles. When based on the principle of geostress variation, the CSUB method is more physically reasonable. However, its limitation lies in the fact that, when solving the diffusion equation for compressible delay interbeds, it does not account for the effects of variations in the discrete nodal cell thickness and hydraulic conductivity of the interbed. This study improves the solution method for the head diffusion equation of compressible delay interbeds based on the principle of geostress variation. The Kozeny-Carman equation was introduced to establish a relationship between the hydraulic conductivity and porosity of the interbeds, while variations in the thickness of discrete nodal cells were also incorporated into the solution process. Collectively, these improvements lead to a more rigorous approach. To verify the effectiveness of the proposed simulation method, three representative test cases were developed and comprehensively compared with the CSUB results. The results indicate that notable discrepancies emerge between the two approaches when the interbed undergoes substantial compression, whereas the method proposed in this study effectively prevents the occurrence of "overcompaction" within the interbed.

Random phase field method for quasi-static and dynamic fracture propagation: Strict phase field equations based on variational principle

A nonlocal variational principle for the converse flexoelectric effect based on simplified strain gradient elasticity

The Universal Variational Principle of Dynamic Complexity: Minimum Effort Transition METP Defined by the 1/e Critical Threshold and Sequential Instability I_seq . A Foundational Law for Predictive Configurational Emergence. V.2

Photoionization time delay in atoms and molecules is a fundamental phenomenon in attosecond physics, encoding essential information about electronic structure and dynamics. Compared with atoms, molecules exhibit anisotropic potentials and additional nuclear degrees of freedom, which make the interpretation of molecular photoionization time delays more intricate but also more informative. In this work, we investigate the dependence of the photoionization time delay on the internuclear distance in the <inline-formula><tex-math id="M3">\begin{document}$ 5\sigma \to k\sigma$\end{document}</tex-math></inline-formula> ionization channel of carbon monoxide (CO) molecules. The molecular ground state is obtained using the Hartree-Fock method, and the photoionization process is treated within quantum scattering theory based on the iterative Schwinger variational principle of the Lippmann–Schwinger equation. Numerical calculations are performed with the ePolyScat program to obtain molecular-frame differential photoionization cross sections and time delays at various internuclear distances. Our results show that the extrema of the photoionization time delay occur near the peaks and dips of the differential cross section and shift toward lower energies as the internuclear distance <i>R</i> increases. At low energies, the time delay along the oxygen end increases with <i>R</i>, while that along the carbon end decreases, which is attributed to the asymmetric charge distribution and the resulting short-range potential difference between the two atomic sites. Around the shape-resonance energy region, both cross section and time delay display pronounced peaks associated with an <inline-formula><tex-math id="M4">\begin{document}$ l=3$\end{document}</tex-math></inline-formula> quasi-bound state. As <i>R</i> increases, the effective potential barrier broadens, the quasi-bound state energy moves to lower values, and its lifetime becomes longer, leading to enhanced resonance amplitude and increased time delay. In the high-energy region, opposite-sign peaks of time delay are found along the O and C directions, corresponding to minima in the cross section. These features are well explained by a two-center interference model, where increasing <i>R</i> shifts the interference minima and the associated time-delay peaks toward lower energies. This study provides deeper insights into the photoionization dynamics of CO molecules, accounting for the role of nuclear motion, and offers valuable references for studying the photoelectron dynamics of more complex molecular systems.

Photoionization time delay in atoms and molecules is a fundamental phenomenon in attosecond physics, encoding essential information about electronic structure and dynamics. Compared with atoms, molecules exhibit anisotropic potentials and additional nuclear degrees of freedom, which make the interpretation of molecular photoionization time delays more intricate but also more informative. In this work, we investigate the dependence of the photoionization time delay on the internuclear distance in the $5\sigma \to k\sigma$ ionization channel of carbon monoxide (CO) molecules. The molecular ground state is obtained using the Hartree–Fock method, and the photoionization process is treated within quantum scattering theory based on the iterative Schwinger variational principle of the Lippmann–Schwinger equation. Numerical calculations are performed with the ePolyScat program to obtain molecular-frame differential photoionization cross sections and time delays at various internuclear distances. Our results show that the extrema of the photoionization time delay occur near the peaks and dips of the differential cross section and shift toward lower energies as the internuclear distance $R$ increases. At low energies, the time delay along the oxygen end increases with $R$, while that along the carbon end decreases, which is attributed to the asymmetric charge distribution and the resulting short-range potential difference between the two atomic sites. Around the shape-resonance energy region, both cross section and time delay display pronounced peaks associated with an $l=3$ quasi-bound state. As $R$ increases, the effective potential barrier broadens, the quasi-bound state energy moves to lower values, and its lifetime becomes longer, leading to enhanced resonance amplitude and increased time delay. In the high-energy region, opposite-sign peaks of time delay are found along the O and C directions, corresponding to minima in the cross section. These features are well explained by a two-center interference model, where increasing $R$ shifts the interference minima and the associated time-delay peaks toward lower energies. This study provides deeper insights into the photoionization dynamics of CO molecules, accounting for the role of nuclear motion, and offers valuable references for studying the photoelectron dynamics of more complex molecular systems.

We revisit the premise that spacetime geometry must be quantized and show that this assumption is not physically required. Just as one does not quantize pressure or temperature, quantizing the metric treats a macroscopic continuum variable as if it were microscopic. We develop an alternative approach, Chronon Field Theory (ChFT), in which a smooth timelike covector Φμ obeys a unified variational principle—the Temporal Coherence Principle (TCP). In appropriate long-wavelength and low-vorticity regimes, the TCP dynamics yield an emergent Lorentzian metric and reproduce the Einstein field equations to leading order. Phase-coherent excitations exhibit a universal invariant speed and admit an eikonal limit that reproduces Hamilton–Jacobi and Schrödinger-type dynamics. Despite the presence of a microscopic causal alignment field, exact operational Lorentz invariance is preserved because all observers and measuring devices co-emerge from the same causal medium. The framework predicts small higher-order dispersive corrections to relativistic propagation while maintaining a universal causal cone, with effects constrained by fast radio burst and multi-messenger observations. ChFT thus provides a compact effective description in which gravitational and quantum dynamics emerge from a single coherence principle, without postulating quantum geometry at the fundamental level.

This study investigates the vibration and buckling of three-layer sandwich nanoplates made from piezoelectric and magnetic materials. This study seeks to determine the fundamental frequencies of four nanosensor combinations made from different materials. This work uses piezoelectric and magnetic materials such PZT5H, BaTiO3, and CoFe2O4 to test nanosensors' electrical and magnetic field sensitivity. The system of equations was generated using Hamilton's variational principle and solved analytically using Navier's method. The findings show that material type and layer topologies significantly affect the natural frequencies of these sensor devices. This study improves multifunctional nanosensor design by analyzing electro-mechanical and magneto-mechanical interactions. .

Thermodynamic variational principle unifying gravity and heat flow

To solve molecular photoionization and electron scattering problems, we use an overset-grid representation of electronic continuum functions, which has an extended central spherical grid that overlaps small spherical grids (subgrids) centered on each atom of a polyatomic molecule. Here, we present an improved algorithm that smoothly partitions the total wave function between the central grid and the atomic subgrids. The smooth partitioning allows one to use approximately one-fourth the number of partial waves on the central grid compared to our previous implementation with switching functions. The resulting numerical method for treating electron scattering and photoionization of polyatomic molecules combines the accuracy and flexibility of pure numerical grid representations with the rapid convergence of hybrid combinations of atom-centered basis-set expansions and grid methods. The overset-grid representation is implemented using the complex Kohn variational principle for scattering and photoionization amplitudes. The faster convergence with respect to the number of central grid partial waves is demonstrated and accuracy is verified by comparisons with the previous implementation and with far more computationally demanding single-center numerical expansions in electron-molecule scattering and photoionization calculations on the neon dimer (Ne2) system, carbon tetrafluoride (CF4) molecule, and the pyridine (C5H5N) molecule in the static-exchange approximation.

This study investigates the free and forced vibrations of an inertial semi-cylindrical cushion using the Lamé displacement–based formulation and the Hamilton-Ostrogradsky variational principle. Natural frequencies, vibration modes, and normal displacements were obtained for different boundary conditions. Special attention was given to the influence of inertial and elastic parameters, including density, Young’s modulus, Poisson’s ratio, and foundation inertia. Analytical relations describing the variation of vibration amplitudes and frequencies with respect to geometric and material characteristics were derived. A comparative analysis between the analytical solution and a 3D finite element model developed in ABAQUS demonstrated strong agreement, confirming the accuracy of the proposed formulation. The results provide practical guidelines for optimizing vibration-isolation elements and improve the design of structures exposed to dynamic loads. The study contributes to Sustainable Development Goal 9 by supporting the development of safer and more durable infrastructure components.

A Variational Principle of the Packing Topological Pressure on Subsets for Non-autonomous Iterated Function Systems

This manuscript introduces the concept of unstable packing topological entropy to quantify the dynamical complexity inherent in arbitrary subsets of partially hyperbolic systems. We develop foundational dimension-theoretic results for this quantity, establishing A distribution principle for unstable packing topological entropy; A variational principle relating the unstable packing topological entropy of an arbitrary compact subset (not necessarily invariant) to the supremum of local unstable packing topological entropies over Borel probability measures supported thereon.

In this paper, we consider the normalized solutions for the following [Formula: see text]-Laplacian critical equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a Lagrange multiplier. Using the concentration compactness lemma, Schwarz rearrangement, Ekeland’s variational principle and minimax theorems, we obtain several existence results under [Formula: see text] and other suitable assumptions. We also analyze the asymptotic behavior of these solutions as [Formula: see text] and [Formula: see text] goes to its upper bound. Moreover, we show the nonexistence result for [Formula: see text] and get that the [Formula: see text]-Laplacian equation has infinitely many solutions by genus theory when [Formula: see text].

Abstract In 2007, Ye & Zhang introduced a version of local topological entropy. Since their entropy function is, as we show under mild conditions, constant for topologically transitive dynamical systems, we propose to adjust the notion in a way that does not neglect the initial transient part of an orbit. We investigate the properties of this “transient” version, which we call translocal entropy, and compute it in terms of Lyapunov exponents for various dynamical systems. We also investigate how this adjustment affects measure-theoretic local (Brin-Katok) entropy and local pressure functions, generalizing some partial variational principles of Ma & Wen.

Active wetting extends classical wetting physics to living systems, in which cells and tissues spread by generating internal forces rather than relying solely on passive interfacial tensions. Unlike passive systems, which evolve toward thermodynamic and mechanical equilibrium by minimizing free energy, active systems remain far from equilibrium due to continuous energy input and dissipation. Their dynamics are sustained, adaptive, and responsive to chemical and mechanical cues in ways that depart fundamentally from passive behavior. In addition, active systems lack a unified energetic or variational principle to describe their evolution. What insights can be drawn from passive models, and how these models might be generalized to account for activity, remain open questions. Studying active wetting may thus reveal new principles of nonequilibrium dynamics at soft and living interfaces, and offer deeper understanding of key biological processes such as wound healing, cancer invasion, and biofilm growth.