Independent Conservation Laws Research Articles

Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton–Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.

Machine learning of independent conservation laws through neural deflation.

We introduce a methodology for seeking conservation laws within a Hamiltonian dynamical system, which we term "neural deflation." Inspired by deflation methods for steady states of dynamical systems, we propose to iteratively train a number of neural networks to minimize a regularized loss function accounting for the necessity of conserved quantities to be in involution and enforcing functional independence thereof consistently in the infinite-sample limit. The method is applied to a series of integrable and nonintegrable lattice differential-difference equations. In the former, the predicted number of conservation laws extensively grows with the number of degrees of freedom, while for the latter, it generically stops at a threshold related to the number of conserved quantities in the system. This data-driven tool could prove valuable in assessing a model's conserved quantities and its potential integrability.

BI-HAMILTONIAN STRUCTURE AND EXACT SOLUTIONS OF ONE BURGERS� TYPE NONLINEAR DYNAMICAL SYSTEM

In the present work, we find the bi-Hamiltonian representation and three classes of ex￾act solutions for the dispersionful (Burgers’ type) nonlinear dynamical system introduced by Szablikowski et al. [13]. In particular, for the above-mentioned system, we construct the infinite hierarchy of functionally independent conservation laws utilizing the gradient holonomic method [3]. Moreover, based on that hierarchy we find the implectic pair of Noetherian operators and corresponding Hamiltonian functionals applying the differential￾algebraic algorithm [8, 12]. Furthermore, we construct three classes of exact traveling wave solutions, in particular, solitary wave and periodic ones, using the –expansion method [18]. It is shown that for the case of the dynamical system under consideration, degrees of the polynomials in cannot be uniquely determined from the system of algebraic equations of the homogeneous balance. Nevertheless, utilizing a more detailed analysis, a general form of the solution is found uniquely. Further, we analyze the obtained results, in particular, the analytical solution is verified by putting it back into original equations. Finally, we anticipate future research objectives, especially finding the standard Lax type representation of the above-mentioned dynamical system.

BI-HAMILTONIAN STRUCTURE AND EXACT SOLUTIONS OF ONE BURGERS� TYPE NONLINEAR DYNAMICAL SYSTEM

In the present work, we find the bi-Hamiltonian representation and three classes of ex￾act solutions for the dispersionful (Burgers’ type) nonlinear dynamical system introduced by Szablikowski et al. [13]. In particular, for the above-mentioned system, we construct the infinite hierarchy of functionally independent conservation laws utilizing the gradient holonomic method [3]. Moreover, based on that hierarchy we find the implectic pair of Noetherian operators and corresponding Hamiltonian functionals applying the differential￾algebraic algorithm [8, 12]. Furthermore, we construct three classes of exact traveling wave solutions, in particular, solitary wave and periodic ones, using the –expansion method [18]. It is shown that for the case of the dynamical system under consideration, degrees of the polynomials in cannot be uniquely determined from the system of algebraic equations of the homogeneous balance. Nevertheless, utilizing a more detailed analysis, a general form of the solution is found uniquely. Further, we analyze the obtained results, in particular, the analytical solution is verified by putting it back into original equations. Finally, we anticipate future research objectives, especially finding the standard Lax type representation of the above-mentioned dynamical system.

Symmetries and conservation laws for a generalization of Kawahara equation

We give a complete description of generalized and formal symmetries for a nonlinear evolution equation which generalizes the Kawahara equation having important applications in the study of plasma waves and capillary–gravity water waves. Using these results and the presence of Hamiltonian structure we also give a complete description of local conservation laws for the equation under study.In particular, we show that the equation in question admits no genuinely generalized symmetries and has only finitely many nontrivial linearly independent local conservation laws, and thus this equation is not symmetry integrable.

The role of quantum coherence in non-equilibrium entropy production

Thermodynamic irreversibility is well characterized by the entropy production arising from non-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a term that only depends on population unbalances, and one that is underpinned by quantum coherences. This allows us to identify a genuine quantum contribution to the entropy production in non-equilibrium quantum processes. We discuss how these features emerge both in Lindblad-Davies differential maps and finite maps subject to the constraints of thermal operations. We also show how this separation naturally leads to two independent entropic conservation laws for the global system-environment dynamics, one referring to the redistribution of populations between system and environment and the other describing how the coherence initially present in the system is distributed into local coherences in the environment and non-local coherences in the system-environment state. Finally, we discuss how the processing of quantum coherences and the incompatibility of non-commuting measurements leads to fundamental limitations in the description of quantum trajectories and fluctuation theorems.

Types of Markov Fields and Tilings

The method of types is one of the most popular techniques in information theory and combinatorics. However, thus far the method has been mostly applied to 1-D Markov processes, and it has not been thoroughly studied for general Markov fields. Markov fields over a finite alphabet of size $m\ge 2$ can be viewed as models for multidimensional systems with local interactions. The locality of these interactions is represented by a shape $\mathcal {S}$ while its marking by symbols of the underlying alphabet is called a tile. Two assignments in a Markov field have the same type if they have the same empirical distribution, i.e., if they have the same number of tiles of a given type. Our goal is to study the growth of the number of possible Markov field types in either a $d$ -dimensional box of lengths $n_{1}, \ldots , n_{d}$ or its cyclic counterpart, a $d$ -dimensional torus. We relate this question to the enumeration of nonnegative integer solutions of a large system of Diophantine linear equations called the conservation laws. We view a Markov type as a vector in a $D=m^{| {\mathcal {S}}|}$ dimensional space and count the number of such vectors satisfying the conservation laws, which turns out to be the number of integer points in a certain polytope. For the torus, this polytope is of dimension $\mu =D-1-\mathrm {rk}( {\mathrm {\mathrm{C}}})$ , where $\mathrm {rk( {\mathrm {\mathrm{C}}})}$ is the number of linearly independent conservation laws $ {\mathrm {\mathrm{C}}}$ . This provides an upper bound on the number of types. Then, we construct a matching lower bound leading to the conclusion that the number of types in the torus Markov field is $\Theta (N^\mu )$ , where $N=n_{1}\ldots n_{d}$ . These results are derived by geometric tools, including ideas of discrete and convex multidimensional geometry.

Adiabaticity and gravity theory independent conservation laws for cosmological perturbations

We carefully study the implications of adiabaticity for the behavior of cosmological perturbations. There are essentially three similar but different definitions of non-adiabaticity: one is appropriate for a thermodynamic fluid $\delta P_{nad}$, another is for a general matter field $\delta P_{c,nad}$, and the last one is valid only on superhorizon scales. The first two definitions coincide if $c_s^2=c_w^2$ where $c_s$ is the propagation speed of the perturbation, while $c_w^2=\dot P/\dot\rho$. Assuming the adiabaticity in the general sense, $\delta P_{c,nad}=0$, we derive a relation between the lapse function in the comoving sli\-cing $A_c$ and $\delta P_{nad}$ valid for arbitrary matter field in any theory of gravity, by using only momentum conservation. The relation implies that as long as $c_s\neq c_w$, the uniform density, comoving and the proper-time slicings coincide approximately for any gravity theory and for any matter field if $\delta P_{nad}=0$ approximately. In the case of general relativity this gives the equivalence between the comoving curvature perturbation $R_c$ and the uniform density curvature perturbation $\zeta$ on superhorizon scales, and their conservation. We then consider an example in which $c_w=c_s$, where $\delta P_{nad}=\delta P_{c,nad}=0$ exactly, but the equivalence between $R_c$ and $\zeta$ no longer holds. Namely we consider the so-called ultra slow-roll inflation. In this case both $R_c$ and $\zeta$ are not conserved. In particular, as for $\zeta$, we find that it is crucial to take into account the next-to-leading order term in $\zeta$'s spatial gradient expansion to show its non-conservation, even on superhorizon scales. This is an example of the fact that adiabaticity (in the thermodynamic sense) is not always enough to ensure the conservation of $R_c$ or $\zeta$.

Some relations between symmetries of nonlocally related systems

The paper is concerned with relations between local symmetries of systems of partial differential equations (PDE) within trees of nonlocally related PDE systems. It is shown that potential systems arising from a given system through linearly independent conservation laws are nonlocally related to each other. Further, a theorem is proven stating that for a PDE system which has precisely n linearly independent local conservation laws, any local symmetry of the PDE system is a projection of some local symmetry of the n-plet potential system. Moreover, a criterion is presented to determine whether or not a specific local symmetry of a given PDE system is a projection of some local symmetry of a specific potential system. Examples are considered. Finally, a formula for a symmetry of a given PDE system in terms of a local symmetry of a nonlocally related subsystem is given. The formula can be used to determine whether a symmetry of the subsystem yields a local or a nonlocal symmetry of the given system, without the need to undertake a full symmetry classification and comparison between the given system and the subsystem.

Functionally independent conservations laws in a quantum integrable model

We study a recently proposed quantum integrable model defined on a lattice with N sites, with fermions or bosons populating each site, as a close relative of the well known spin- Gaudin model. This model has 2N arbitrary parameters, a linear dependence on an interaction type parameter x, and can be solved exactly. It has N known constants of motion that are linear in x. We display further constants of motion with higher fermion content that are linearly independent of the known conservation laws. Our main result is that despite the existence of the higher conservation laws, the model has only N functionally independent conservation laws. Therefore we propose that N can be viewed as the number of degrees of freedom, in parallel to the classical definition of integrability.

Preclusion of switch behavior in networks with mass-action kinetics

We study networks taken with mass-action kinetics and provide a Jacobian criterion that applies to an arbitrary network to preclude the existence of multiple positive steady states within any stoichiometric class for any choice of rate constants. We are concerned with the characterization of injective networks, that is, networks for which the species formation rate function is injective in the interior of the positive orthant within each stoichiometric class. We show that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish. The function consists of components of the species formation rate function and a maximal set of independent conservation laws. The determinant of the function is a polynomial in the species concentrations and the rate constants (linear in the latter) and its coefficients are fully determined. The criterion also precludes the existence of degenerate steady states. Further, we relate injectivity of a network to that of the network obtained by adding outflow, or degradation, reactions for all species.

Variable elimination in post-translational modification reaction networks with mass-action kinetics

We define a subclass of chemical reaction networks called post-translational modification systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop a mathematical framework based on the notion of a cut (a particular subset of species in the system), which provides a linear elimination procedure to reduce the number of variables in the system to a set of core variables. The steady states are parameterized algebraically by the core variables, and graphical conditions for when steady states with positive core variables imply positivity of all variables are given. Further, minimal cuts are the connected components of the species graph and provide conservation laws. A criterion for when a (maximal) set of independent conservation laws can be derived from cuts is given.

A normal form of completely integrable systems

The purpose of this article is to show that a C1 differential system on Rn which admits a set of n−1 independent C2 conservation laws defined on an open subset Ω⊆Rn, is essentially C1 equivalent on an open and dense subset of Ω, with the linear differential system u1′=u1,u2′=u2,…,un′=un. The main results are illustrated in the case of two concrete dynamical systems, namely the three dimensional Lotka–Volterra system, and respectively the Euler equations from the free rigid body dynamics.

Alfvén solitons in the coupled derivative nonlinear Schrödinger system with symbolic computation

The propagation of nonlinear Alfvén waves in magnetized plasmas with right and left circular polarizations is governed by the coupled derivative nonlinear Schrödinger (CDNLS) system. The integrability of this system is indicated by the existence of two gauge-equivalent Lax pairs and infinitely many independent conservation laws. With symbolic computation, the analytic one- and two-soliton solutions are obtained via the Hirota bilinear method. The propagation characteristics of the Alfvén waves are discussed through qualitative analysis. The collision dynamics of the CDNLS solitons is found to be characterized by the invariance of the soliton velocities and widths, parameter-dependent changes of the soliton amplitudes and conservation of the total energy of right- and left-polarized components. The parametric condition for the amplitude-preserving collision occurring in each component is explicitly given.

Conservation laws of multidimensional diffusion--convection equations

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations u t=(A(u)) ii +(B i(u)) i were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O.~Popovych, N.M. Ivanova, J. Math. Phys. 46, 2005, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of conservation laws are constructed.

Polynomial Conservation Laws in Quantum Systems

We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e., the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.

Conservation laws in non-homogeneous electro-magneto-elastic materials

Noether's theorem on invariant variational principles is applied for linear transversely isotropic non-homogeneous electro-magneto-elastic material. Invariant condition of the electromagnetic enthalpy is derived and expressed in original manner. Usually, inserting infinitesimal symmetry-transformations relating to material space of homogeneous materials into the invariant condition, one can obtain some expressions for the non-homogeneous material. These expressions indicate the relations between material inhomogeneity force and energy-momentum tensor. In order to obtain conservation laws of the non-homogeneous material, it follows from the invariant condition that the material coefficients have to satisfy a set of first-order linear partial differential equations with which there is a possible mathematical form of infinitesimal symmetry-transformation relating to material space. Satisfaction of these partial differential equations not only assures the existence of infinitesimal symmetry-transformations but also determines the number of them. Several independent and non-trivial conservation laws relating to material space with different non-homogeneous material coefficients are given. Some results of the path-independent integrals emanating from the conservation laws calculated around a crack tip are presented.

Steady-state vectorial self-diffraction on a non-local photorefractive grating in a crystal of symmetry \overline{4} 3 m at symmetrical transmitting geometry

The steady-state two-wave interaction in a cubic crystal of the symmetry group \(\)3m with the non-local photorefractive response in the absence of an external electric field is considered for the case of arbitrary interaction orientation with respect to the crystallographic coordinate system and for arbitrary intensities and polarization states of incident light waves. The self-diffraction problem is described on the basis of four coupled-wave equations in terms of the complex scalar amplitudes of components of the light waves with orthogonal linear polarization. The derived conservation laws are valid for the non-linear dependency of the photorefractive-grating amplitude on the modulation coefficient of the interference light pattern. It follows from these laws that the two non-unidirectional energy fluxes can form the total energy exchange between the two interacting light waves. A set of independent conservation laws allows us to decouple the coupled-wave equations and to obtain their analytical solution, at least, in the form of quadrature formulae. For example, such a solution is derived for the case of linearly polarized incident light waves and for the linearized dependency of the photorefractive-grating amplitude on the modulation coefficient. The explicit analytical expressions for the scalar amplitudes are obtained for the transversal electro-optic configuration of interaction. The possibility of polarization-state transformation of light waves without energy exchange between them is shown.

Conservation laws in the one-dimensional Hubbard model

We examine the nature, number, and interrelation of conservation laws in the one-dimensional Hubbard model. In previous work by Shastry [Phys. Rev. Lett. 56, 1529 (1986); 56, 2334 (1986); 56, 2453 (1986); J. Stat. Phys. 50, 57 (1988)], who studied the model on a large but finite number of lattice sites (N{sub a}), only N{sub a}+1 conservation laws, corresponding to N{sub a}+1 operators that commute with themselves and the Hamiltonian, were explicitly identified, rather than the {approx}2N{sub a} conservation laws expected from the solvability and integrability of the model. Using a pseudoparticle approach related to the thermodynamic Bethe ansatz, we discover an additional N{sub a}+1 independent conservation laws corresponding to nonlocal, mututally commuting operators, which we call transfer-matrix currents. Further, for the model defined in the whole Hilbert space, we find there are two other independent commuting operators (the squares of the {eta}-spin and spin operators) so that the total number of local plus nonlocal commuting conservation laws for the one-dimensional Hubbard model is 2N{sub a}+4. Finally, we introduce an alternative set of 2N{sub a}+4 conservation laws which assume particularly simple forms in terms of the pseudoparticle and Yang-particle operators. This set of mutually commuting operators lends itself more readilymore » to calculations of physically relevant correlation functions at finite energy or frequency than the previous set.« less

Classification of local conservation laws of Maxwell's equations

A complete and explicit classification of all independent local conservation laws of Maxwell's equations in four dimensional Minkowski space is given. Besides the elementary linear conservation laws, and the well-known quadratic conservation laws associated to the conserved stress-energy and zilch tensors, there are also chiral quadratic conservation laws which are associated to a new conserved tensor. The chiral conservation laws possess odd parity under the electric–magnetic duality transformation of Maxwell's equations, in contrast to the even parity of the stress-energy and zilch conservation laws. The main result of the classification establishes that every local conservation law of Maxwell's equations is equivalent to a linear combination of the elementary conservation laws, the stress-energy and zilch conservation laws, the chiral conservation laws, and their higher order extensions obtained by replacing the electromagnetic field tensor by its repeated Lie derivatives with respect to the conformal Killing vectors on Minkowski space. The classification is based on spinorial methods and provides a direct, unified characterization of the conservation laws in terms of Killing spinors.