Showing papers on "Conservation law published in 2016"

Stability and Boundary Stabilization of 1-D Hyperbolic Systems

Abstract: This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

Causes of Irregular Energy Density in $f(R,T)$ Gravity

Abstract: We investigate irregularity factors for a self-gravitating spherical star evolving in the presence of an imperfect fluid. We explore the gravitational field equations and the dynamical equations with the systematic construction in $f(R,T)$ gravity, where $T$ is the trace of the energy-momentum tensor. Furthermore, we analyze two well-known differential equations (which occupy principal importance in the exploration of causes of energy density inhomogeneities) with the help of the Weyl tensor and the conservation laws. The irregularity factors for a spherical star are examined for particular cases of dust and isotropic and anisotropic fluids in dissipative and nondissipative regimes in the framework of $f(R,T)$ gravity. It is found that, as the complexity of the matter with the anisotropic stresses increases, the inhomogeneity factor corresponds more closely to one of the structure scalars.

Nonequilibrium quantum dynamics and transport: from integrability to many-body localization

Abstract: We review the non-equilibrium dynamics of many-body quantum systems after a quantum quench with spatial inhomogeneities, either in the Hamiltonian or in the initial state. We focus on integrable and many-body localized systems that fail to self-thermalize in isolation and for which the standard hydrodynamical picture breaks down. The emphasis is on universal dynamics, non-equilibrium steady states and new dynamical phases of matter, and on phase transitions far from thermal equilibrium. We describe how the infinite number of conservation laws of integrable and many-body localized systems lead to complex non-equilibrium states beyond the traditional dogma of statistical mechanics.

Integrable nonlocal nonlinear equations

Abstract: A nonlocal nonlinear Schrodinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$ symmetric thus the nonlocal NLS equation is also $PT$ symmetric. In this paper, new {\it reverse space-time} and {\it reverse time} nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space-time, and in some cases reverse time, nonlocal nonlinear Schrodinger, modified Korteweg-deVries (mKdV), sine-Gordon, $(1+1)$ and $(2+1)$ dimensional three-wave interaction, derivative NLS, "loop soliton", Davey-Stewartson (DS), partially $PT$ symmetric DS and partially reverse space-time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space-time and reverse time nonlocal discrete nonlinear Schrodinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painleve type equations are derived from the reverse space-time and reverse time nonlocal NLS equations.

Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy

Abstract: We discuss the physical interpretation of stress-energy tensors that source static spherically symmetric Kerr-Schild metrics. We find that the sources of such metrics with no curvature singularities or horizons do not simultaneously satisfy the weak and strong energy conditions. Sensible stress-energy tensors usually satisfy both of them. Under most circumstances, these sources are not perfect fluids and contain shear stresses. We show that for these systems the classical double copy associates the electric charge density to the Komar energy density. In addition, we demonstrate that the stress-energy tensors are determined by the electric charge density and their conservation equations.

Magnetic Corrections to the Soft Photon Theorem

Abstract: The soft photon theorem, in its standard form, requires corrections when the asymptotic particle states carry magnetic charges. These corrections are deduced using electromagnetic duality and the resulting soft formula conjectured to be exact for all Abelian gauge theories. Recent work has shown that the standard soft theorem implies an infinity of conserved electric charges. The associated symmetries are identified as "large" electric gauge transformations. Here the magnetic corrections to the soft theorem are shown to imply a second infinity of conserved magnetic charges. The associated symmetries are identified as large magnetic gauge transformations. The large magnetic symmetries are naturally subsumed in a complexification of the electric ones.

Tail effect in gravitational radiation reaction: Time nonlocality and renormalization group evolution

Abstract: We use the effective field theory (EFT) framework to calculate the tail effect in gravitational radiation reaction, which enters at the fourth post-Newtonian order in the dynamics of a binary system. The computation entails a subtle interplay between the near (or potential) and far (or radiation) zones. In particular, we find that the tail contribution to the effective action is nonlocal in time and features both a dissipative and a “conservative” term. The latter includes a logarithmic ultraviolet (UV) divergence, which we show cancels against an infrared (IR) singularity found in the (conservative) near zone. The origin of this behavior in the long-distance EFT is due to the point-particle limit—shrinking the binary to a point—which transforms a would-be infrared singularity into an ultraviolet divergence. This is a common occurrence in an EFT approach, which furthermore allows us to use renormalization group (RG) techniques to resum the resulting logarithmic contributions. We then derive the RG evolution for the binding potential and total mass/energy, and find agreement with the results obtained imposing the conservation of the (pseudo) stress-energy tensor in the radiation theory. While the calculation of the leading tail contribution to the effective action involves only one diagram, five are needed for the one-point function. This suggests logarithmic corrections may be easier to incorporate in this fashion. We conclude with a few remarks on the nature of these IR/UV singularities, the (lack of) ambiguities recently discussed in the literature, and the completeness of the analytic post-Newtonian framework.

Constructing higher-order hydrodynamics: The third order.

Abstract: Hydrodynamics can be formulated as the gradient expansion of conserved currents in terms of the fundamental fields describing the near-equilibrium fluid flow. In the relativistic case, the Navier-Stokes equations follow from the conservation of the stress-energy tensor to first order in derivatives. In this paper, we go beyond the presently understood second-order hydrodynamics and discuss the systematization of obtaining the hydrodynamic expansion to an arbitrarily high order. As an example of the algorithm that we present, we fully classify the gradient expansion at third order for neutral fluids in four dimensions, thus finding the most general next-to-leading-order corrections to the relativistic Navier-Stokes equations in curved space-time. In doing so, we list 20 new transport coefficient candidates in the conformal case and 68 in the nonconformal case. As we do not consider any constraints that could potentially arise from the local entropy current analysis, this is the maximal possible set of neutral third-order transport coefficients. To investigate the physical implications of these new transport coefficients, we obtain the third-order corrections to the linear dispersion relations that describe the propagation of diffusion and sound waves in relativistic fluids. We also compute the corrections to the scalar (spin-2) two-point correlation function of the third-order stress-energy tensor. Furthermore, as an example of a nonlinear hydrodynamic flow, we calculate the third-order corrections to the energy density of a boost-invariant Bjorken flow. Finally, we apply our field theoretic results to the $\mathcal{N}=4$ supersymmetric Yang-Mills fluid at infinite 't Hooft coupling and an infinite number of colors to find the values of five new linear combinations of the conformal transport coefficients.

Well-posedness of a conservation law with non-local flux arising in traffic flow modeling

Abstract: We prove the well-posedness of entropy weak solutions of a scalar conservation law with non-local flux arising in traffic flow modeling. The result is obtained providing accurate $$\mathbf {L^\infty }$$L?, BV and $$\mathbf {L^1}$$L1 estimates for the sequence of approximate solutions constructed by an adapted Lax-Friedrichs scheme.

A Two-Stage Fourth Order Time-Accurate Discretization for Lax--Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws

Abstract: In this paper we develop a novel two-stage fourth order time-accurate discretization for time-dependent flow problems, particularly for hyperbolic conservation laws. Different from the classical Runge--Kutta (R--K) temporal discretization for first order Riemann solvers as building blocks, the current approach is solely associated with Lax--Wendroff (L--W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using the well-developed four-stage procedure for R--K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L--W type schemes for the construction of a two-stage fourth order scheme.

The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability

Abstract: The three-dimensional (3-D) nonlinear and dispersive PDEs system for surface waves propagating at undisturbed water surface under the gravity force and surface tension effects are studied. By applying the reductive perturbation method, we derive the (2 + 1) -dimensions form of the Davey-Stewartson (DS) system for the modulation of 2-D harmonic waves. By using the simplest equation method, we find exact traveling wave solutions and a general form of the multiple-soliton solution of the DS model. The dispersion analysis as well as the conservation law of the DS system are discussed. It is revealed that the consistency of the results with the conservation of the potential energy increases with increasing Ursell parameter. Also, the stability of the ODEs form of the DS system is presented by using the phase portrait method.

Beyond heat baths: Generalized resource theories for small-scale thermodynamics

Abstract: Thermodynamics has recently been extended to small scales with resource theories that model heat exchanges. Real physical systems exchange diverse quantities: heat, particles, angular momentum, etc. We generalize thermodynamic resource theories to exchanges of observables other than heat, to baths other than heat baths, and to free energies other than the Helmholtz free energy. These generalizations are illustrated with "grand-potential" theories that model movements of heat and particles. Free operations include unitaries that conserve energy and particle number. From this conservation law and from resource-theory principles, the grand-canonical form of the free states is derived. States are shown to form a quasiorder characterized by free operations, d majorization, the hypothesis-testing entropy, and rescaled Lorenz curves. We calculate the work distillable from-and we bound the work cost of creating-a state. These work quantities can differ but converge to the grand potential in the thermodynamic limit. Extending thermodynamic resource theories beyond heat baths, we open diverse realistic systems to modeling with one-shot statistical mechanics. Prospective applications such as electrochemical batteries are hoped to bridge one-shot theory to experiments.

Hydromechanical Modeling of Unsaturated Flow in Double Porosity Media

Abstract: Geomaterials with aggregated structure or containing fissures often exhibit a bimodal pore size distribution that can be viewed as two coexisting pore regions of different scales. The double-porosity concept enables continuum modeling of such materials by considering two interacting pore scales satisfying relevant conservation laws. This paper develops a thermodynamically consistent framework for hydromechanical modeling of unsaturated flow in double-porosity media. With an explicit treatment of the two pore scales, conservation laws are formulated incorporating an effective stress tensor that is energy-conjugate to the rate of deformation tensor of the solid matrix. A constitutive framework is developed on the basis of energy-conjugate pairs identified in the first law of thermodynamics, which is then incorporated into a three-field mixed finite-element formulation for double-porosity media. Numerical simulations of laboratory- and field-scale problems are presented to demonstrate the impact of d...

Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow

Abstract: The local fractional Burgers’ equation (LFBE) is investigated from the point of view of local fractional conservation laws envisaging a nonlinear local fractional transport equation with a linear non-differentiable diffusion term. The local fractional derivative transformations and the LFBE conversion to a linear local fractional diffusion equation are analyzed.

A Floquet model for the many-body localization transition

Abstract: The nature of the dynamical quantum phase transition between the many-body localized (MBL) phase and the thermal phase remains an open question, and one line of attack on this problem is to explore this transition numerically in finite-size systems. To maximize the contrast between the MBL phase and the thermal phase in such finite-size systems, we argue one should choose a Floquet model with no local conservation laws and rapid thermalization to ``infinite temperature'' in the thermal phase. Here we introduce and explore such a Floquet spin chain model and show that standard diagnostics of the MBL-to-thermal transition behave well in this model even at modest sizes. We also introduce a physically motivated space-time correlation function, which peaks at the transition in the Floquet model, but is strongly affected by conservation laws in Hamiltonian models.

On the Muskat problem: Global in time results in 2D and 3D

Abstract: This paper considers the three-dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the available estimates to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in spaces with critical regularity, giving solutions with bounded slope and time integrable bounded curvature.

A note on generalized hydrodynamics: inhomogeneous fields and other concepts

Abstract: Generalized hydrodynamics (GHD) was proposed recently as a formulation of hydrodynamics for integrable systems, taking into account infinitely-many conservation laws. In this note we further develop the theory in various directions. By extending GHD to all commuting flows of the integrable model, we provide a full description of how to take into account weakly varying force fields, temperature fields and other inhomogeneous external fields within GHD. We expect this can be used, for instance, to characterize the non-equilibrium dynamics of one-dimensional Bose gases in trap potentials. We further show how the equations of state at the core of GHD follow from the continuity relation for entropy, and we show how to recover Euler-like equations and discuss possible viscosity terms.

Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation

Abstract: Under investigation in this paper is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the water wave interaction. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study the integrability of the equation, including its bilinear representation, soliton solutions, periodic wave solutions, Backlund transformation and Lax pairs, respectively. Furthermore, by virtue of its Lax equations, the infinite conservation laws of the equation are also derived with the recursion formulas. Finally, the asymptotic behavior of periodic wave solutions is shown with a limiting procedure.

Three-Dimensional MHD Flow of Casson Fluid in Porous Medium with Heat Generation

Abstract: The magnetohydrodynamic (MHD) three-dimensional boundary layer flow of an incompressible Casson fluid in a porous medium is investigated. Heat transfer characteristics are analyzed in the presence of heat generation/absorption. Laws of conservation of mass, momentum and energy are utilized. Results are computed and analyzed for the velocities, temperature, skin-friction coefficients and local Nusselt number.

Exact steady states for quantum quenches in integrable Heisenberg spin chains

Abstract: What is the long-time behavior of a quantum system after it is brought out of equilibrium? In a generic case, it has been proposed that an effective thermalization occurs, and the local properties of the system can then be described by a thermal Gibbs ensemble. However, in integrable systems, with infinitely many conservation laws, the answer is far more complicated. It was shown in several special cases that the long-time steady state can be exactly described by the so called complete generalized Gibbs ensemble. The latter can be obtained by maximizing the entropy with constraints imposed by the conservation of energy, but also by additional carefully chosen subset of the integrals of motion. In this work, the authors analyze more general initial states, focusing on integrable XXZ Heisenberg spin chains. They find that the complete generalized Gibbs ensemble indeed correctly describes the system at long times after the quench.

$f(R)$-gravity from Killing Tensors

Abstract: We consider $f\left(R\right) $-gravity in a Friedmann-Lemaitre-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of $f\left(R\right) $ and the field equations to be invariant under Lie-Backlund transformations which are linear in the momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether's Theorem. We find three new integrable $f\left(R\right) $ models, for which with the application of the conservation laws we reduce the field equations to a system of two first-order ordinary differential equations. For each model we study the evolution of the cosmological fluid. Where we find that for the one integrable model the cosmological fluid has an equation of state parameter, in which in the latter there is a linear behavior in terms of the scale factor which describes the Chevallier, Polarski and Linder (CPL) parametric dark energy model.

Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles

Abstract: A generalized (3+1)-dimensional nonlinear wave is investigated, which describes many nonlinear phenomena in liquid containing gas bubbles. In this paper, a lucid and systematic approach is proposed to systematically study the complete integrability of the equation by using Bell’s polynomials scheme. Its bilinear equation, N-soliton solution and Backlund transformation with explicit formulas are successfully structured, which can be reduced to the analogues of (3+1)-dimensional KP equation, (3+1)-dimensional nonlinear wave equation and Korteweg-de Vries equation, respectively. Moreover, the infinite conservation laws of the equation are found by using its Backlund transformation. All conserved densities and fluxes are presented with explicit recursion formulas. Furthermore, by employing Riemann theta function, the one- and two-periodic wave solutions for the equation are constructed well. Finally, an asymptotic relation is presented, which implies that the periodic wave solutions can be degenerated to the soliton solutions under some special conditions.

Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada–Kotera–Ito equation

Abstract: In this paper Lie symmetry analysis of the seventh-order time fractional Sawada–Kotera–Ito (FSKI) equation with Riemann–Liouville derivative is performed. Using the Lie point symmetries of FSKI equation, it is shown that it can be transformed into a nonlinear ordinary differential equation of fractional order with a new dependent variable. In the reduced equation the derivative is in Erdelyi–Kober sense. Furthermore, adapting the Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we obtain conservation laws of the underlying equation. In addition, we construct some exact travelling wave solutions for the FSKI equation using the sub-equation method.

BMS invariance and the membrane paradigm

Abstract: The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat spacetime. It is infinite dimensional and entails an infinite number of conservation laws. According to the black hole membrane paradigm, null infinity (in asymptotically flat spacetime) and black hole event horizons behave like fluid membranes. The fluid dynamics of the membrane is governed by an infinite set of symmetries and conservation laws. Our main result is to point out that the infinite set of symmetries and conserved charges of the BMS group and the membrane paradigm are the same. This relationship has several consequences. First, it sheds light on the physical interpretation of BMS conservation laws. Second, it generalizes the BMS conservation laws to arbitrary subregions of arbitrary null surfaces. Third, it clarifies the identification of the superrotation subgroup of the BMS group. We briefly comment on the black hole information problem.