arXiv subjects

Shouhong Wang

Publications and source records attributed to Shouhong Wang.

At least 19 recordsLinked to original sources

arXiv2025

Cahn-Hilliard Equations on Lattices: Dynamic Transitions and Pattern Formations

This article examines the dynamic phase transitions and pattern formations attributed to binary systems modeled by the Cahn-Hilliard equation. In particular, we consider a two-dimensional lattice structure and determine how different choices of the spanning vectors influence the resulting dynamical tramsitions and pattern formations. As the basic steady-state loses its linear stability, the binary system undergoes a dynamic transition which is shown to be characterized by both the geometry of the domain and the choice of physical parameters of the model. Unlike rectangular domains, we are able to observe the emergence of hexagonally-packed circles, as well as the familiar rolls and square structures. We begin with the decomposition of our function space into a stable and unstable eigenspace before calculating the center manifold that maps the former to the later. In analyzing the resulting reduced equations, we consider the different multiplicities that the critical eigenvalue can have, which is shown to be geometry-dependent. We briefly consider the long-range interaction model and determine that it produces similar results to the original model.

math.AP↗

arXiv2022

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that two modes become critical and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to $S^3$. The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

physics.ao-ph↗

arXiv2022

Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns. However, in presence of random fluctuations, this reduction procedure is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds and energy estimates measuring the defect of parameterization of the high-modes. To operate for fluid problems, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes that suitably counterbalance the loss of regularity due to the nonlinear terms. The approach enables us to predict, from the reduced equations, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly. Our parameterization formulas involve non-Markovian coefficients, which depend explicitly on the history of the noise path that drives the SPDE dynamics, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-Benard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.

math.AP↗

arXiv2018

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criteria for nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength $Λ$ of the basic flow crosses a critical threshold $Λ_c$. Also we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition parameter $A$, fully characterizing the nonlinear interactions of different modes. A systematic numerical method is carried out to explore transition in different flow parameter regimes. We find that the system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$. Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, continuous and catastrophic transitions to spatiotemporal oscillations.

physics.flu-dyn↗

arXiv2017

Baroclinic Instability and Transitions in a Two-Layer Quasigeostrophic Channel Model

The main objective of this article is to derive a mathematical theory associated with the nonlinear stability and dynamic transitions of the basic shear flows associated with baroclinic instability, which plays a fundamental role in the dominant mechanism shaping the cyclones and anticyclones that dominate weather in mid-latitudes, as well as the mesoscale eddies that play various roles in oceanic dynamics and the transport of tracers. This article provides a general transition and stability theory for the two-layer quasi-geostrophic model originally derived by Pedlosky \cite{ped1970}. We show that the instability and dynamic transition of the basic shear flow occur only when the Froude number $F>(γ^2+1)/2$, where $γ$ is the weave number of lowest zonal harmonic. In this case, we derive a precise critical shear velocity $U_c$ and a dynamic transition number $b$, and we show that the dynamic transition and associated flo0w patterns are dictated by the sign of this transition number. In particular, we show that for $b>0$, the system undergoes a continuous transition, leading to spatiotemporal flow oscillatory patterns as the shear velocity $U$ crosses $U_c$; for $b<0$, the system undergoes a jump transition, leading to drastic change and more complex transition patterns in the far field, and to unstable period solutions for $U<U_c$. Furthermore, we show that when the wavenumber of the zonal harmonic $γ=1$, the transition number $b$ is always positive and the system always undergoes a continuous dynamic transition leading spatiotemporal oscillations. This suggests that a continuous transition to spatiotemporal patterns is preferable for the shear flow associated with baroclinic instability.

physics.ao-ph↗

arXiv2015

Low-Dimensional Galerkin Approximations of Nonlinear Delay Differential Equations

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

nlin.CD↗

arXiv2015

Dynamic Transitions of Quasi-Geostrophic Channel Flow

The main aim of this paper is to describe the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In \cite{CGSW03}, the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend the results in \cite{CGSW03} by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a non-dimensional number $γ$ which controls the transition behavior. We prove that depending on $γ$, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of $γ$ for a physically realistic region of parameter space suggest that a catastrophic transition is preferred in this flow.

physics.ao-ph↗

arXiv2014

Unified Field Equations Coupling Four Forces and Principle of Interaction Dynamics

The main objective of this article is to postulate a principle of interaction dynamics (PID) and to derive unified field equations coupling the four fundamental interactions based on first principles. PID is a least action principle subject to div$_A$-free constraints for the variational element with $A$ being gauge potentials. The Lagrangian action is uniquely determined by 1) the principle of general relativity, 2) the $U(1)$, $SU(2)$ and $SU(3)$ gauge invariances, 3) the Lorentz invariance, and 4) principle of principle of representation invariance (PRI), introduced in [11]. The unified field equations are then derived using PID. The unified field model spontaneously breaks the gauge symmetries, and gives rise to a new mechanism for energy and mass generation. The unified field model introduces a natural duality between the mediators and their dual mediators, and can be easily decoupled to study each individual interaction when other interactions are negligible. The unified field model, together with PRI and PID applied to individual interactions, provides clear explanations and solutions to a number of outstanding challenges in physics and cosmology, including e.g. the dark energy and dark matter phenomena, the quark confinement, asymptotic freedom, short-range nature of both strong and weak interactions, decay mechanism of sub-atomic particles, baryon asymmetry, and the solar neutrino problem.

physics.gen-ph↗

arXiv2013

Non-Markovian Reduced Systems for Stochastic Partial Differential Equations: The Additive Noise Case

This article proposes for stochastic partial differential equations (SPDEs) driven by additive noise, a novel approach for the approximate parameterizations of the ``small'' scales by the ``large'' ones, along with the derivaton of the corresponding reduced systems. This is accomplished by seeking for stochastic parameterizing manifolds (PMs) introduced in a previous work by the authors(*), which are random manifolds aiming to provide --- in a mean square sense --- such approximate parameterizations. Backward-forward systems are designed to give access to such PMs as pullback limits depending through the nonlinear terms on the time-history of the dynamics of the low modes when the latter is simply approximated by its stochastic linear component. It is shown that the corresponding pullback limits can be efficiently determined, leading in turn to an operational procedure for the derivation of non-Markovian reduced systems able to achieve good modeling performances in practice. This is illustrated on a stochastic Burgers-type equation, where it is shown that the corresponding non-Markovian features of these reduced systems play a key role to reach such performances.

math.AP↗

arXiv2013

On stochastic parameterizing manifolds: Pullback characterization and Non-Markovian reduced equations

A general approach to provide approximate parameterizations of the "small" scales by the "large" ones, is developed for stochastic partial differential equations driven by linear multiplicative noise. This is accomplished via the concept of parameterizing manifolds (PMs) that are stochastic manifolds which improve in mean square error the partial knowledge of the full SPDE solution $u$ when compared to the projection of $u$ onto the resolved modes, for a given realization of the noise. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers (as parameterized by the sought PM) as a pullback limit depending on the time-history of the modes with low wave numbers. The resulting manifolds obtained by such a procedure are not subject to a spectral gap condition such as encountered in the classical theory. Instead, certain PMs can be determined under weaker non-resonance conditions. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. Such reduced systems take the form of SDEs involving random coefficients that convey memory effects via the history of the Wiener process, and arise from the nonlinear interactions between the low modes, embedded in the "noise bath." These random coefficients follow typically non-Gaussian statistics and exhibit an exponential decay of correlations whose rate depends explicitly on gaps arising in the non-resonances conditions. It is shown on a stochastic Burgers-type equation, that such PM-based reduced systems can achieve very good performance in reproducing statistical features of the SPDE dynamics projected onto the resolved modes, such as the autocorrelations and probability functions of the corresponding modes amplitude.

math.AP↗

arXiv2013

Unified Field Theory and Principle of Representation Invariance

This article consists of two parts. The main objectives of Part 1 are to postulate a new principle of representation invariance (PRI), and to refine the unified field model of four interactions, derived using the principle of interaction dynamics (PID). Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint, and PRI requires that physical laws be independent of representations of the gauge groups. One important outcome of this field model is a natural duality between the interacting fields and the adjoint bosonic fields. This duality predicts two Higgs particles of similar mass with one due to weak interaction and the other due to strong interaction. The field model can be naturally decoupled to study individual interactions, leading to 1) modified Einstein equations, giving rise to a unified theory for dark matter and dark energy, 2) three levels of strong interaction potentials for quark, nucleon/hadron, and atom respectively, and 3) two weak interaction potentials. These potential/force formulas offer a clear mechanism for both quark confinement and asymptotic freedom. The main objectives of Part 2 are 1) to propose a sub-leptons and sub-quark model, which we call weakton model, and 2) to derive a mechanism for all sub-atomic decays and bremsstrahlung. The weakton model postulates that all matter particles and mediators are made up of massless weaktons. The weakton model offers a perfect explanation for all sub-atomic decays and all generation/annihilation precesses of matter-antimatter. In particular, the precise constituents of particles involved in all decays both before and after the reaction can now be precisely derived. In addition, the bremsstrahlung phenomena can be understood using the weakton model. Also, the weakton model offers an explanation to the baryon asymmetry problem.

physics.gen-ph↗

arXiv2013

Pattern Formations of 2D Rayleigh-Bènard Convection with No-Slip Boundary Conditions for the Velocity at the Critical Length Scales

We study the Rayleigh-B{é}nard convection in a 2-D rectangular domain with no-slip boundary conditions for the velocity. The main mathematical challenge is due to the no-slip boundary conditions, since the separation of variables for the linear eigenvalue problem which works in the free-slip case is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold $R_c$, the system bifurcates to an attractor, which is an $(m-1)$--dimensional sphere, where $m$ is the number of eigenvalues which cross zero as R crosses $R_c$. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when $m=2$. More precisely, we rigorously prove that when $m=2$, the bifurcated attractor is homeomorphic to a one-dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers.

nlin.PS↗

arXiv2012

Gravitational Field Equations and Theory of Dark Matter and Dark Energy

The main objective of this article is to derive a new set of gravitational field equations and to establish a new unified theory for dark energy and dark matter. The new gravitational field equations with scalar potential $φ$ are derived using the Einstein-Hilbert functional, and the scalar potential $φ$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{ij}$, the scalar potential $φ$ and their interactions, unified by the new gravitational field equations. Associated with the scalar potential $φ$ is the scalar potential energy density $\frac{c^4}{8πG} Φ=\frac{c^4}{8πG} g^{ij}D_iD_j φ$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M ΦdM=0$. The sum of this new potential energy density $\frac{c^4}{8πG} Φ$ and the coupling energy between the energy-momentum tensor $T_{ij}$ and the scalar potential field $φ$ gives rise to a new unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8πG}{c^4} T + Φ$. Furthermore, the new field equations resolve a few difficulties encountered by the classical Einstein field equations.

physics.gen-ph↗

arXiv2012

Dynamic Transition and Pattern Formation for Chemotactic Systems

The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.

nlin.AO↗

arXiv2011

Remarks on the Rayleigh-Benard Convection on Spherical Shells

The main objective of this article is to study the effect of spherical geometry on dynamic transitions and pattern formation for the Rayleigh-Benard convection. The study is mainly motivated by the importance of spherical geometry and convection in geophysical flows. It is shown in particular that the system always undergoes a continuous (Type-I) transition to a $2l_c$-dimensional sphere $S^{2lc}$, where lc is the critical wave length corresponding to the critical Rayleigh number. Furthermore, it has shown in [12] that it is critical to add nonisotropic turbulent friction terms in the momentum equation to capture the large-scale atmospheric and oceanic circulation patterns. We show in particular that the system with turbulent friction terms added undergoes the same type of dynamic transition, and obtain an explicit formula linking the critical wave number (pattern selection), the aspect ratio, and the ratio between the horizontal and vertical turbulent friction coefficients.

math-ph↗

arXiv2011

Tropical Atmospheric Circulations with Humidity Effects

The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modeling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity and the humidity dependent heat source is modeled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain the same types of transitions and hence the scenario of the El Niño mechanism as described in \cite{MW2,MW3}. The effect of the moisture only lowers slightly the magnitude of the critical thermal Rayleigh number.

physics.ao-ph↗

arXiv2011

Pattern Formation in Rayleigh Benard Convection

The main objective of this article is to study the three-dimensional Rayleigh-Benard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m-1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m=1, the structure of this attractor is trivial. When m=2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m=2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.

nlin.PS↗

arXiv2011

Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility

The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equation, following the ideas of the dynamic transition theory developed recently by Ma and Wang.

math-ph↗