Zhang Equation Research Articles

On the compatibility of a family of generalized Keller–Segel models with the Kardar–Parisi–Zhang equation: A traveling wave study

In this paper, we prove the existence of (semi-)bounded traveling wave solutions to a family of generalized Keller–Segel models with nonlinear production–degradation mechanism, closely related to the Schrödinger–Doebner–Goldin equation through the Madelung’s hydrodynamic formulation (as shown in J. L. López, A quantum approach to Keller–Segel dynamics via a dissipative nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst. Ser. A 41 (2021) 2601–2617), in which the evolution of the chemical concentration is governed by a Kardar–Parisi–Zhang-type equation. The main idea behind it consists in demonstrating the existence of a heteroclinic orbit in the traveling variable to the underlying dynamical system, through the detailed construction of an invariant region and the subsequent use of the Poincaré–Bendixson theorem. When possible, explicit traveling waves (or implicit relations involving them) are also constructed, along with the specific form of the source terms giving rise to them.

Jointly invariant measures for the Kardar–Parisi–Zhang equation

Jointly invariant measures for the Kardar–Parisi–Zhang equation

High moments of the SHE in the clustering regimes

High moments of the SHE in the clustering regimes

Impact of Fenton aging on the incipient motion of microplastic particles in open-channel flow

Impact of Fenton aging on the incipient motion of microplastic particles in open-channel flow

Developing a semi-analytical model for estimating mechanical properties of sandstone reservoirs: Enhancing applications in hydrocarbon production and underground gas storage

Developing a semi-analytical model for estimating mechanical properties of sandstone reservoirs: Enhancing applications in hydrocarbon production and underground gas storage

Boundary current fluctuations for the half‐space ASEP and six‐vertex model

AbstractWe study fluctuations of the current at the boundary for the half‐space asymmetric simple exclusion process (ASEP) and the height function of the half‐space six‐vertex model at the boundary at large times. We establish a phase transition depending on the effective density of particles at the boundary, with Gaussian symplectic ensemble (GSE) and Gaussian orthogonal ensemble (GOE) limits as well as the Baik–Rains crossover distribution near the critical point. This was previously known for half‐space last‐passage percolation, and recently established for the half‐space log‐gamma polymer and Kardar–Parisi–Zhang equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half‐space six‐vertex model and a half‐space Hall–Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half‐space ASEP and six‐vertex model, indicating a hidden free fermionic structure.

Space–time fluctuation of the Kardar–Parisi–Zhang equation in d≥3 and the Gaussian free field

Space–time fluctuation of the Kardar–Parisi–Zhang equation in d≥3 and the Gaussian free field

Comparison of Soil Hydraulic Properties Estimated by Steady- and Unsteady-Flow Methods in the Laboratory

In this study, soil hydraulic conductivity (K) and soil sorptivity (S) values estimated by applying various steady- and unsteady-flow methods using cumulative infiltration data of three disturbed soils (sandy loam, loam, clay) obtained from a disc infiltrometer in the laboratory at various negative pressure heads were compared. The steady-flow methods used were those of Ankeny et al. and Reynolds and Elrick as well as Logsdon and Jaynes, while the unsteady-flow methods were those of Haverkamp et al. (two-term (2T) and three-term (3T) infiltration equations) and Zhang. The method of White et al., which is a steady-flow method but also uses unsteady-flow infiltration data, was also examined. The results showed that the three steady-flow methods, as well as the Zhang equation, for values of the van Genuchten coefficient n > 1.35, tend to give similar values of K. The 2T infiltration equation with β = 0.6 provided hydraulic conductivity values greater than those estimated by the steady-state methods but gave negative K values in some cases. The values of the coefficients C1 and C2 of the 2T equation were affected by the infiltration time. The coefficient C1 increased while C2 decreased with increasing time when the cumulative linearization method (CL) was applied, but the change in C1 tended to be smaller than that in C2. The inverse solution of the 3T equation using the Excel Solver application for β = 0.75 and β = 1.6, when positive values of K were obtained, approached better the K values estimated by the steady-flow methods compared with those estimated using β = 0.6. Regarding the estimation of S from the unsteady-flow equations (2T, 3T, Zhang), comparable S values were obtained by all equations. The differences between the S values of the various methods are smaller compared to those of K, and S is less affected than K in terms of time. The problem of negative estimates of K might be attributed to the fact that the soils used in this study are classified as soils situated in the domain of lateral capillarity or are not completely homogeneous or soil compaction is observed at some depth. In the case where the soils are not completely homogeneous, the Sequential Infiltration Analysis (SIA) method with β = 0.75 corresponding to the soil types studied was proved to be effective in estimating K values.

Field-Theoretic Renormalization Group in Models of Growth Processes, Surface Roughening and Non-Linear Diffusion in Random Environment: Mobilis in Mobili

This paper is concerned with intriguing possibilities for non-conventional critical behavior that arise when a nearly critical strongly non-equilibrium system is subjected to chaotic or turbulent motion of the environment. We briefly explain the connection between the critical behavior theory and the quantum field theory that allows the application of the powerful methods of the latter to the study of stochastic systems. Then, we use the results of our recent research to illustrate several interesting effects of turbulent environment on the non-equilibrium critical behavior. Specifically, we couple the Kazantsev–Kraichnan “rapid-change” velocity ensemble that describes the environment to the three different stochastic models: the Kardar–Parisi–Zhang equation with time-independent random noise for randomly growing surface, the Hwa–Kardar model of a “running sandpile” and the generalized Pavlik model of non-linear diffusion with infinite number of coupling constants. Using field-theoretic renormalization group analysis, we show that the effect can be quite significant leading to the emergence of induced non-linearity or making the original anisotropic scaling appear only through certain “dimensional transmutation”.

Experimental study of underwater explosions below a free surface: Bubble dynamics and pressure wave emission

The current work experimentally studies the complex interaction between underwater explosion (UNDEX) bubbles and a free surface. We aim to reveal the dependence of the associated physics on the key factor, namely, the dimensionless detonation depth γ (scaled by the maximum equivalent bubble radius). Four typical bubble behavior patterns are identified with the respective range of γ: (i) bubble bursting at the free surface, (ii) bubble jetting downward, (iii) neutral collapse of the bubble, and (iv) quasi-free-field motion. By comparison of the jet direction and the migration of the bubble centroid, a critical value of γ is vital for ignoring the effects of the free surface on UNDEX bubbles. Good agreements are obtained between the experimental data and Zhang equation [Zhang et al., “A unified theory for bubble dynamics,” Phys. Fluids 35, 033323 (2023)]. Additionally, the dependence of the pressure signals in the flow field on γ is investigated. The peak pressure, impulse, and energy dissipation in the UNDEX are investigated.

Exact solutions of Wu–Zhang equation via complete discrimination system for polynomial method

In this paper, the complete discrimination system for the polynomial method is applied to solve the Wu–Zhang system, and all the possible exact solutions are obtained, these exact solutions can be applied to the exploration of nonlinear physical phenomena, the method in this paper is different from the existing literature studies on the Wu–Zhang equation. By taking different parameters, interesting graphs are plotted for all the obtained solutions. The results confirm that the proposed method is effective and can be used to solve a variety of nonlinear consistency time fractional partial differential equations.

Improved finite-difference and pseudospectral schemes for the Kardar–Parisi–Zhang equation with long-range temporal correlations

Improved finite-difference and pseudospectral schemes for the Kardar–Parisi–Zhang equation with long-range temporal correlations

Stationary measures of the KPZ equation on an interval from Enaud–Derrida’s matrix product ansatz representation

The stationary measures of the Kardar–Parisi–Zhang equation on an interval have been computed recently. We present a rather direct derivation of this result by taking the weak asymmetry limit of the matrix product ansatz for the asymmetric simple exclusion process. We rely on the matrix product ansatz representation of Enaud and Derrida, which allows to express the steady-state in terms of re-weighted simple random walks. In the continuum limit, its measure becomes a path integral (or re-weighted Brownian motion) of the form encountered in Liouville quantum mechanics, recovering the recent formula.

Symplectic Geometry Aspects of the Parametrically-Dependent Kardar–Parisi–Zhang Equation of Spin Glasses Theory, Its Integrability and Related Thermodynamic Stability

A thermodynamically unstable spin glass growth model described by means of the parametrically-dependent Kardar-Parisi-Zhang equation is analyzed within the symplectic geometry-based gradient-holonomic and optimal control motivated algorithms. The finitely-parametric functional extensions of the model are studied, and the existence of conservation laws and the related Hamiltonian structure is stated. A relationship of the Kardar-Parisi-Zhang equation to a so called dark type class of integrable dynamical systems, on functional manifolds with hidden symmetries, is stated.

A stationary model of non-intersecting directed polymers

We consider the partition function of non-intersecting continuous directed polymers of length t in dimension , in a white noise environment, starting from positions and terminating at positions . When , it is well known that for fixed x, the field solves the Kardar–Parisi–Zhang equation and admits the Brownian motion as a stationary measure. In particular, as t goes to infinity, converges to the exponential of a Brownian motion B(y). In this article, we show an analogue of this result for any . We show that converges as t goes to infinity to an explicit functional of independent Brownian motions. This functional admits a simple description as the partition sum for non-intersecting semi-discrete polymers on lines. We discuss applications to the endpoints and midpoints distribution for long non-crossing polymers and derive explicit formula in the case of two polymers. To obtain these results, we show that the stationary measure of the O’Connell–Warren multilayer stochastic heat equation is given by a collection of independent Brownian motions. This in turn is shown via analogous results in a discrete setup for the so-called log-gamma polymer and exploit the connection between non-intersecting log-gamma polymers and the geometric Robinson–Schensted–Knuth correspondence found in Corwin-O’Connell-Seppäläinen-Zygouras (2014 Duke Math. J. 163 513–63).

Sediment transport capacity equation for soils from the Loess Plateau and northeast China

Sediment transport capacity equation for soils from the Loess Plateau and northeast China

Symmetries and Zero Modes in Sample Path Large Deviations

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin–Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar–Parisi–Zhang equation with flat initial profile.

A poroelasticity theory for soil incorporating adsorption and capillarity

Adsorption and capillarity, in the order of high free energy to low, are the two soil–water interaction mechanisms controlling the hydro-mechanical behaviour of soils. Yet most of the poroelasticity theories of soil are based on capillarity only, leading to misrepresentations of hydro-mechanical behaviour in the low free energy regime beyond vaporisation. This inability is reasoned to be caused by two major limitations in the existing theories: missing interparticle attraction energy and incomplete definition of adsorption-induced pore-water pressure. A poroelasticity theory is formulated to incorporate the two soil–water interaction mechanisms, and the transition between them – that is, condensation/vaporisation, by expanding the classical three-phase mixture system to a four-phase mixture system with adsorptive water as an additional phase. An interparticle attractive stress is identified as one of the key sources for deformation and strength of soils induced by adsorption and is implemented in the poroelasticity theory. A recent breakthrough concept of soil sorptive potential is utilised to establish the physical link between adsorption-induced pore-water pressure and matric suction. The proposed poroelasticity theory can be reduced to several previous theories when interparticle attractive stress is ignored. The new theory is used to derive the effective stress equation for variably saturated soil by identifying energy-conjugated pairs. The derived effective stress equation leads to Zhang and Lu's unified effective stress equation, and can be reduced to Bishop's effective stress equation when only the capillary mechanism is considered and to Terzaghi's effective stress equation when a saturated condition is imposed. The derived effective stress equation is experimentally validated for a variety of soil in the full matric suction range, substantiating the validity and accuracy of the poroelasticity theory for soil under variably saturated conditions.

Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ

Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ

An analytical approach to solve the fractional‐order (2 + 1)‐dimensional Wu–Zhang equation

This paper considers time‐fractional ‐dimensional Wu–Zhang nonlinear system of partial differential equation describing a long dispersive wave. An approximate analytical solution of the dispersion relation of the long wave has been obtained by the fractional reduced differential transform method (FRDTM). The effect of fractional‐order on the wave profile of the solution is discussed graphically and comparing the exact solution of Wu–Zhang equation when . The result shows that the present method reveals the effectiveness, efficiency, and reliability of computed mathematical results to easily solve the fractional‐order Wu–Zhang (WZ) system of differential equations.