Nontrivial Conservation Law Research Articles

Nonlinear dynamics of an epidemic compartment model with asymptomatic infections and mitigation

A significant proportion of the infections driving the current SARS-CoV-2 pandemic are transmitted asymptomatically. Here we introduce and study a simple epidemic model with separate compartments comprising asymptomatic and symptomatic infected individuals. The linear dynamics determining the outbreak condition of the model is equivalent to a renewal theory approach with exponential waiting time distributions. Exploiting a nontrivial conservation law of the full nonlinear dynamics, we derive analytic bounds on the peak number of infections in the absence and presence of mitigation through isolation and testing. The bounds are compared to numerical solutions of the differential equations.

A modified formal Lagrangian formulation for general differential equations

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler--Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether's Theorem and in particular, a nontrivial conservation law of the Fornberg--Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.

Addendum: Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics (2018 J. Phys. Commun. 2 115018)

An extension is proposed to the internal symmetry transformations associated with mass, entropy and other Clebsch-related conservation in geophysical fluid dynamics. Those symmetry transformations were previously parameterized with an arbitrary function of materially conserved Clebsch potentials. The extension consists in adding potential vorticity q to the list of fields on which a new arbitrary function depends. If , where is an arbitrary function of specific entropy s, then the symmetry is trivial and gives rise to a trivial conservation law. Otherwise, the symmetry is non-trivial and an associated non-trivial conservation law exists. Moreover, the notions of trivial and non-trivial Casimir invariants are defined. All non-trivial symmetries that become hidden following a reduction of phase space are associated with non-trivial Casimir invariants of a non-canonical Hamiltonian formulation for fluids, while all trivial conservation laws are associated with trivial Casimir invariants.

Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.

A new Painlevé integrable Broer-Kaup system: symmetry analysis, analytic solutions and conservation laws

Purpose This study aims to find the symmetries and conservation laws of a new Painlevé integrable Broer-Kaup (BK) system with variable coefficients. This system is an extension of dispersive long wave equations. As the system is generalized and new, it is essential to explore some of its possible aspects such as conservation laws, symmetries, Painleve integrability, etc. Design/methodology/approach This paper opted for an exploratory study of a new Painleve integrable BK system with variable coefficients. Some analytic solutions are obtained by Lie classical method. Then the conservation laws are derived by multiplier method. Findings This paper presents a complete set of point symmetries without any restrictions on choices of coefficients, which subsequently yield analytic solutions of the series and solitary waves. Next, the authors derive every admitted non-trivial conservation law that emerges from multipliers. Research limitations/implications The authors have found that the considered system is likely to be integrable. So some other aspects such as Lax pair integrability, solitonic behavior and Backlund transformation can be analyzed to check the complete integrability further. Practical implications The authors develop a time-dependent Painleve integrable long water wave system. The model represents more specific data than the constant system. The authors presented analytic solutions and conservation laws. Originality/value The new time-dependent Painleve integrable long water wave system features some interesting results on symmetries and conservation laws.

To Conserve, or Not to Conserve: A Review of Nonconservative Theories of Gravity

Apart from the familiar structure firmly-rooted in the general relativistic field equations where the energy–momentum tensor has a null divergence i.e., it conserves, there exists a considerable number of extended theories of gravity allowing departures from the usual conservative framework. Many of these theories became popular in the last few years, aiming to describe the phenomenology behind dark matter and dark energy. However, within these scenarios, it is common to see attempts to preserve the conservative property of the energy–momentum tensor. Most of the time, it is done by means of some additional constraint that ensures the validity of the standard conservation law, as long as this option is available in the theory. However, if no such extra constraint is available, the theory will inevitably carry a non-trivial conservation law as part of its structure. In this work, we review some of such proposals discussing the theoretical construction leading to the non-conservation of the energy–momentum tensor.

Gauge symmetries and isometries in higher dimensions

We study the invariance properties of five-dimensional metrics and their corresponding geodesic equations of motion. In this context a number of five-dimensional models of the Einstein–Gauss–Bonnet (EGB) theory leading to black holes, wormholes and spacetime horns arising in a variety of situations are discussed in the context of variational symmetries of which each vector field, via Noether’s theorem (NT), provides a nontrivial conservation law. In particular, it is shown that algebraic structure of isometries and the variational conservation laws of the five-dimensional Einstein–Bonnet metric extend consistently from the well-known Minkowski, de-Sitter and Schwarzschild four-dimensional spacetimes to the considered five-dimensional ones. In the equivalent five-dimensional case, the maximal algebra of kvs is fifteen with eight additional Noether symmetries. Also, whereas the constant curvature five-dimensional case leads to fifteen kvs and one additional Noether symmetry and seven plus one in the minimal case, a number of metrics of the EGB theory in five dimensions give rise to algebras isomorphic a seven-dimensional algebra of kvs and a single additional Noether symmetry.

On the different types of global and local conservation laws for partial differential equations in three spatial dimensions: Review and recent developments

For systems of partial differential equations in three spatial dimensions, dynamical conservation laws holding on volumes, surfaces, and curves, as well as topological conservation laws holding on surfaces and curves, are studied in a unified framework. Both global and local formulations of these different conservation laws are discussed, including the forms of global constants of motion. The main results consist of providing an explicit characterization for when two conservation laws are locally or globally equivalent, and for when a conservation law is locally or globally trivial, as well as deriving relationships among the different types of conservation laws. In particular, the notion of a “trivial” conservation law is clarified for all of the types of conservation laws. Moreover, as further new results, conditions under which a trivial local conservation law on a domain can yield a non-trivial global conservation law on the domain boundary are determined and shown to be related to differential identities that hold for PDE systems containing both evolution equations and spatial constraint equations. Numerous physical examples from fluid flow, gas dynamics, electromagnetism, and magnetohydrodynamics are used as illustrations.

Geometry and conservation laws for a class of second-order parabolic equations I: Geometry

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate G-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quantities invariant up to a generalized change of variables). One family of invariants gives a geometric characterization for parabolic equations of Monge–Ampère type. A second family of invariants determines when a parabolic equation has a local choice of coordinates putting it in evolutionary form.In addition to their intrinsic interest, these results are applied in a follow up paper on the conservation laws of parabolic equations. It is shown there that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge–Ampère type.

Escape of a harmonically forced particle from an infinite-range potential well: a transient resonance

The paper considers a process of escape of classical particle from a one-dimensional potential well by virtue of an external harmonic forcing. We address a particular model of the infinite-range potential well that allows independent adjustment of the well depth and of the frequency of small oscillations. The problem can be conveniently reformulated in terms of action-angle variables. Further averaging provides a nontrivial conservation law for the slow flow. Thus, one can consider the problem in terms of averaged dynamics on primary 1:1 resonance manifold. This simplification allows efficient analytic exploration of the escape process, and yields a theoretical prediction for minimal forcing amplitude required for the escape, as a function of the excitation frequency. This function exhibits a single minimum for certain intermediate frequency value. Numeric simulations are in complete qualitative and reasonable quantitative agreement with the theoretical predictions.

Noether symmetries and stability of ideal gas solutions in Galileon cosmology

A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noether's Theorem to determine conservation laws for the field equations. In the Friedmann-Lema\^itre-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

On conservation laws for a viscous fluid

The equation of motion of viscous fluid have the divergence type of the conservation laws and admit the infinite Lie algebra of point symmetries. Canonical operators of the algebra give rise to conservation laws of the first order. This was tested for the basic operators of admitted algebra. Any conservation law may be turned by doing some canonical operator on the nontrivial conservation law. The canonical operators of the zero order of conservation laws were calculated. The problem about the calculation of canonical operators of the first order of conservation laws are supplied.

Self-adjointness and conservation laws of a generalized Burgers equation

A (2 + 1)-dimensional generalized Burgers equation is considered. Having written this equation as a system of two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law.

Emergence of fractal behavior in condensation-driven aggregation

We investigate the condensation-driven aggregation model that we recently proposed whereby an initial ensemble of chemically identical Brownian particles are continuously growing by condensation and at the same time undergo aggregation upon collision. We solved the model exactly by using scaling theory for the case when a particle, say of size x , grows by an amount alphax over the time it takes to collide with another particle of any size. It is shown that the particle size spectra exhibit transition to scaling c(x,t) approximately t;{-beta}varphi(xt{z}) accompanied by the emergence of a fractal of dimension d {f}=1/(1+2alpha) . A remarkable feature of this model is that it is governed by a nontrivial conservation law, namely, the d {f}th moment of c(x,t) is time invariant. The reason why it remains conserved is explained. Exact values for the exponents beta , z , and d{f} are obtained and it is shown that they obey a generalized scaling relation beta=(1+d{f})z .

Condensation-driven aggregation in one dimension

We propose a model for aggregation where particles are continuously growing by heterogeneous condensation in one dimension, and solve it exactly. We show that the particle size spectra exhibit a transition to dynamic scaling c(x,t) approximately t-beta phi(x/tz) . The exponents beta and z satisfy a generalized scaling relation beta=(1+q)z where the value of q is fixed by a nontrivial conservation law. We show that the value of 1+q is always less than the value 2 for aggregation without condensation.

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-Kahler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.

Geometry of conservation laws for a class of parabolic partial differential equations

I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system \({\cal I}\) on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of \({\cal I}\) modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem.Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then \({\cal I}\) has a deprolongation to an equivalent system \({\cal J}\) on a 7-manifold N, and any conservation law for \({\cal I}\) can be expressed as a closed 3-form on N that lies in \({\cal J}\). Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (uxxuyy-u2xy)+Buxx+2Cuxy+Duyy+E = 0 where A, B, C, D, E are functions of x, y, t, u, ux, uy, ut. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation \( u_t = {1 \over 2} e^{-u} (u_{xx}+u_{yy}) \) has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.

THE N = 3 SUPERSYMMETRIC KdV HIERARCHY

An N = 3 supersymmetric extension of the KdV equation, whose Hamiltonian structure is the classical O(3)-extended superconformal algebra is presented. Integrability of the equation is argued based on the existence of a non-trivial conservation law which reduces to the first non-trivial conservation law of the KdV equation.