Conservation laws in acoustics

The method earlier proposed by the author for obtaining a complete set of conservation laws is applied to a number of simple acoustic problems. Some of the conservation laws presented are derived for the first time. Special attention is given to the physical interpretation of the results.     

ADM pseudotensors,conserved quantities and covariant conservation laws in general relativity

The ADM formalism is reviewed and techniques for decomposing generic components of metric, connection and curvature are obtained. These techniques will turn out to be enough to decompose not only Einstein equations but also covariant conservation laws. Then a number of independent sets of hypotheses that are sufficient (though not necessary) to obtain standard ADM quantities (and Hamiltonian) from covariant conservation laws are considered. This determines explicitly the range in which standard techniques are equivalent to covariant conserved quantities.The Schwarzschild metric in different coordinates is then considered, showing how the standard ADM quantities fail dramatically in non-Cartesian coordinates or even worse when asymptotically flatness is not manifest; while, in view of their covariance, covariant conservation laws give the correct result in all cases.     

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.     

Relation of permutation symmetry of the tensor of nonlinear susceptibility to the conservation laws of photon number in nonlinear quadratic nondissipative and partially dissipative media

Relation between permutation symmetry for components of the tensor of the nonlinear quadratic susceptibility and the conservation laws of the number of photons in a nondissipative nonlinear quadratic medium is pointed out in this work. On the basis of validity of partial conservation laws of the number of photons some relations of the permutation symmetry for components of the tensor of the nonlinear quadratic susceptibility are derived even in the case of a partially dissipative medium.     

Conservative quantum computing

The Wigner-Araki-Yanase theorem shows that conservation laws limit the accuracy of measurement. Here, we generalize the argument to show that conservation laws limit the accuracy of quantum logic operations. A rigorous lower bound is obtained of the error probability of any physical realization of the controlled-NOT gate under the constraint that the computational basis is represented by a component of spin, and that physical implementations obey the angular momentum conservation law. The lower bound is shown to be inversely proportional to the number of ancilla qubits or the strength of the external control field.     

On conservation laws and boundary conditions for “short waves” equation

In this paper we study local conservation laws for the equation of short waves in the form of a variational problem. We analyze an infinite symmetry group of the equation and generate a finite number of conservation laws corresponding to given infinite symmetries through appropriate boundary conditions.     

Conservation laws derived by the Neutral-Action Method

Conservation laws are a recognized tool in physical- and engineering sciences. The classical procedure to construct conservation laws is to apply Noether's Theorem. It requires the existence of a Lagrange-function for the system under consideration. Two unknown sets of functions have to be found. A broader class of such laws is obtainable, if Noether's Theorem is used together with the Bessel-Hagen extension, raising the number of sets of unknown functions to three. By using the recently developed Neutral-Action Method, the same conservation laws can be obtained by calculating only one unknown set of functions. Moreover the Neutral Action Method can also be applied in the absence of a Lagrangian, since only the governing differential equations are required for this procedure. In the paper, an application of this method to the Schr?dinger equation is presented.     

NOETHER'S Theorem and Higher Conservation Laws in Ultrashort Pulse Propagation

LAMB has constructed a number of local conservation laws for the differential equation σ_τ = sin σ describing ultrashort pulse propagation in a resonant medium. On the other hand it is known that from any solution of this equation others may be found by BÄCKLUND transformation. In this paper we introduce extended BÄCKLUND transformations B_a and compose them to form infinitesimal invariance transformations B_a+ε B from which via NOETHER'S theorem a system of conservation laws follows. Some of them are recognized as LAMB'S conservation laws. Another conservation law results from a scale invariance called LIE'S transformation.     

Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

We show how one can construct conservation laws of the Liang equation which is not variational but may be regarded as Euler-Lagrange in part. This first requires the determination of the Noether-type symmetries associated with the partial Lagrangian. The final construction of the conservation laws resort to a formula equivalent to Noether’s theorem. A variety of subclasses are given and, for each, a large number of conserved flows are found—the method is usable for any general choice of the variable speed of sound.     

Stability in bihamiltonian systems and multidimensional rigid body

The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is “simpler”. Thus it is natural to ask: can the dynamics of a bihamiltonian system be understood by studying its hamiltonian pair, without studying the associated first integrals?     

A general theory of conservation laws,their violation,and spontaneous phenomena

We formulate a general theory of conservation laws and other invariants for a physical system through equivalence relations. The conservation laws are classified according to the type of equivalence relation, with group equivalence, homotopical equivalence, and other types of equivalence relations giving respective kinds of conservation laws. The stability properties in the topological (and differentiable) sense are discussed using continuous deformations with respect to control parameters. The conservation laws due to the Abelian symmetries are shown to be stable through application of well-known theorems.     

Inference of analytical thermodynamic models for biological networks

We present an automated algorithm for inferring analytical models of closed reactive biochemical mixtures, on the basis of standard approaches borrowed from thermodynamics and kinetic theory of gases. As input, the method requires a number of steady states (i.e. an equilibria cloud in phase–space), and at least one time series of measurements for each species. Validations are discussed for both the Michaelis–Menten mechanism (four species, two conservation laws) and the mitogen-activated protein kinase–MAPK mechanism (eleven species, three conservation laws).     

A note on the interplay between symmetries,reduction and conservation laws of Stokes’ first problem for third-grade rotating fluids

We investigate the invariance properties, nontrivial conservation laws and interplay between these notions that underly the equations governing Stokes’ first problem for third-grade rotating fluids. We show that a knowledge of this leads to a number of different reductions of the governing equations and, thus, a number of exact solutions can be obtained and a spectrum of further analyses may be pursued.     

Infinitely-many conservation laws for two (2+1)-dimensional nonlinear evolution equations in fluids

In this paper, a method that can be used to construct the infinitely-many conservation laws with the Lax pair is generalized from the (1+1)-dimensional nonlinear evolution equations (NLEEs) to the (2+1)-dimensional ones. Besides, we apply that method to the Kadomtsev–Petviashvili (KP) and Davey–Stewartson equations in fluids, and respectively obtain their infinitely-many conservation laws with symbolic computation. Based on that method, we can also construct the infinitely-many conservation laws for other multidimensional NLEEs possessing the Lax pairs, including the cylindrical KP, modified KP and (2+1)-dimensional Gardner equations, in fluids, plasmas, optical fibres and Bose–Einstein condensates.     

Bifurcation of Nonclassical Viscous Shock Profiles from the Constant State

We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic field, nonclassical waves involve two fields in an essential way. This feature is reflected in the viscous profile differential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurcation occurring in the classical case. We carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points. One consequence of our analysis is that weak transitional waves arise in pairs, with profiles forming a 2-cycle configuration previously shown to lead to nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws, rather than with change of type in the associated hyperbolic system.     

Conservation Laws of a Discrete Soliton System

In this paper, we obtain an infinite number of conservation laws for a discrete soliton system by using a solvable generalized Riccati equation.     

Ordered hadron S-Matrix

An ordered hadron S-matrix is developed that accommodates an arbitrary number of “neighbors” for each particle. Automatic features are baryon-number conservation and zero triality. The ordered Hilbert space splits into a set of non-communicating sectors each characterized by a “skeleton” graph whose external edges can be given aquark interpretation. Selection rules are found that generalize the OZI rules.     

Conservation laws for a vortex in a compressible nonequilibrium medium

The problem of conservation of magnitudes is considered for a vortex in a relaxing compressible medium. Heat release due to the relaxation of a nonequilibrium medium leads to the propagation of compression waves, which remove material. Traditional integrals of motion are inapplicable in this case. We pro-pose the concept of integral quantity, which is conserved with an arbitrary degree of accuracy despite the fact that waves cross the boundary of the integration domain. Based on this concept, a broad class of conservation laws is derived for axisymmetric disturbances of columnar vortices, including conservation of the circulation and total angular momentum of the vortex. For nonaxisymmetric disturbances, it is shown that the total angular momentum and properly defined energy integral are conserved. Numerical verification of the derived conservation laws is performed and the perspectives for using these conservation laws in numerical simulations are discussed.     

Onsager Relations and Eulerian Hydrodynamic Limit for Systems with Several Conservation Laws

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsager's reciprocity relations must hold. We check validity of these identities whenever a stationary measure with product structure exists. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena. The Onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel. The fact that equilibrium thermodynamic entropy is Lax entropy for the arising Euler equations was noticed earlier in the context of Hamiltonian systems with weak noise, see ref. 7.