Hierarchy Of Conservation Laws Research Articles

On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

On the Integrability of Lattice Equations with Two Continuum Limits

We study a new example of a lattice equation, which is one of the key equations of a generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits, which are the well-known fifth-order partial-differential equations, namely, the Sawada–Kotera and Kaup-Kupershmidt equations. We justify its integrability by constructing an L-A pair and a hierarchy of conservation laws.

The Novel Lie-Algebraic Approach to Studying Integrable Heavenly Type Multi-Dimensional Dynamical Systems

The review is devoted to a novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems and its relationships to old and recent investigations of the classical Buhl problem of describing compatible linear vector field equations, its general Pfeiffer and modern Lax-Sato type special solutions. Eespecially we analyze the related Lie-algebra structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Pleban′ski and later analyzed in a series of articles. The AKS-algebraic and related A¯ÂÂ’-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie–Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly type equations under consideration. It is shown that all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a very interesting Lagrange–d’Alembert type mechanical interpretation of the devised integrability scheme with the Lax–Sato equations is also discussed.

Conservation laws of shallow water theory and the Galilean relativity principle

By the example of a model of shallow water theory, it is shown that the compatibility analysis of the Hugoniot conditions for various basic systems of conservation laws in the coordinate system moving together with a strong discontinuity can lead to erroneous results. It is connected with the hierarchy of conservation laws in shallow water theory with respect to the Galilean transformation, according to which the conservation law for total energy is unconditionally noninvariant with respect to this transformation, which leads to the dependence of the corresponding Hugoniot condition on the velocity of the inertial reference frame. It is shown that the specified shortcoming of the classical shallow water theory is absent in the model of vortex shallow water suggested by V. M. Teshukov.

Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner-Nordstrom spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface crossing the future event horizon. We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon. A hierarchy of conservations laws on degenerate horizons is also derived.

On the hierarchy of conservation laws in a cellular automaton

Conservation laws in cellular automata (CA) are studied as an abstraction of the conservation laws observed in nature. In addition to the usual real-valued conservation laws we also consider more general group-valued and semigroup-valued conservation laws. The (algebraic) conservation laws in a CA form a hierarchy, based on the range of the interactions they take into account. The conservation laws with smaller interaction ranges are the homomorphic images of those with larger interaction ranges, and for each specific range there is a most general law that incorporates all those with that range. For one-dimensional CA, such a most general conservation law has—even in the semigroup-valued case—an effectively constructible finite presentation, while for higher-dimensional CA such effective construction exists only in the group-valued case. It is even undecidable whether a given two-dimensional CA conserves a given semigroup-valued energy assignment. Although the local properties of this hierarchy are tractable in the one-dimensional case, its global properties turn out to be undecidable. In particular, we prove that it is undecidable whether this hierarchy is trivial or unbounded. We point out some interconnections between the structure of this hierarchy and the dynamical properties of the CA. In particular, we show that positively expansive CA do not have non-trivial real-valued conservation laws.

A Geometric Study of the Dispersionless Boussinesq Type Equation

We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney–Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in [4]. The results include: A description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each other but are neither Hamiltonian, nor symplectic. These operators give rise to a noncommutative infinite-dimensional algebra of recursion operators.

(Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to theN= 1 supersymmetric KdV equation

Using methods of Kersten et al (2004 J. Geom. Phys. 50 273–302) and Krasil'shchik and Kersten (2000 Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Dordrecht: Kluwer)), we accomplish an extensive study of the N = 1 supersymmetric Korteweg–de Vries (KdV) equation. The results include a description of local and nonlocal Hamiltonian and symplectic structures, five hierarchies of symmetries, the corresponding hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws. We stress that the main point of the paper is not just the results on super-KdV equation itself, but merely exposition of the efficiency of the geometrical approach and of the computational algorithms based on it.

Unitized Jordan algebras and dispersionless KdV equations

Multicomponent KdV systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied. Recursion laws for conserved densities are given under the assumption that the algebra possesses a unity element. Sufficient conditions are given for the unitized counterpart of a diagonalizable non-unital system to be diagonalizable. Hamiltonian structure is discussed within the context of DN Jordan algebras and N scattering problems.

Strong field limit of the Born-Infeldp-form electrodynamics

We study the strong field limit of p-form Born-Infeld theory. It turns out that this limiting theory is a unique theory displaying the full symmetry group of the underlying canonical structure. Moreover, being a nonlinear theory, it possesses an infinite hierarchy of conservation laws.

Jordan Manifolds and Dispersionless KdV Equations

Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied, under the assumptions that the Jordan algebra has a unity element and a compatible non-degenerate inner product. Much of this structure may be encoded in a so-called Jordan manifold, akin to a Frobenius manifold. In particular the Hamiltonian properties of these systems are investigated.

Hamiltonian Structure of Benney Type Hydrodynamic and Boltzmann-Vlasov Kinetic Equations on an Axis and Some Applications to Manufacturing Science

Some years ago Zakharov and Gibbon observed a very nice relation between the Benney type equation in hydrodynamics and the Vlasov equation of kinetic theory. These equations are generalized and put into the framework of infinite-dimensional Lie algebras associated to Lie algebra structures on rings of functions on finite-dimensional manifolds. This gives rise to a complete description of the Hamiltonian structure of both types of equations under consideration. In particular, their Lax type representations together with an infinite involutive hierarchy of conservation laws are obtained in an exact form. Some applications to chaotic manyparticle dynamical systems, turbulent fluid flows and swept volume analysis are considered.

Abelian current algebra and the Virasoro algebra on the lattice

We describe how a natural lattice analogue of the abelian current algebra combined with free discrete time dynamics gives rise to the lattice Virasoro algebra and a corresponding hierarchy of conservation laws.

Non-Abelian pseudopotentials and conservation laws of reaction-diffusion equations

The authors apply the Wahlquist-Estabrook technique to reaction-diffusion equations. A representation for non-Abelian pseudopotentials is obtained. An infinite hierarchy of conservation laws is derived from these pseudopotentials. The class of equations associated with such pseudopotentials is determined and examples are presented.

SUPERSYMMETRY AND CONSERVATION LAWS IN THE KdV SYSTEM

The role of supersymmetry in the KdV-MKdV hierarchy of conservation laws is examined. Among all the fifth order evolution equations, the Lax equation turns out to be the only one which is mapped onto its modified partner through the pair of supersymmetric transformations.

New Conservation Laws for Linear Quasi-Geostrophic Waves in Shear

Abstract There exists an infinite set of quadratic conserved quantities for linear quasi-geostrophic waves in horizontal and vertical shear, the first two members of the set corresponding to the pseudomomentum and pseudo-energy conservation laws that lead to the Rayleigh-Kuo (or Charney-Stern) and the Fjortoft stability criteria. This infinite hierarchy of conservation laws follows from the conservation of the pseudomomentum in each eigenmode of the shear flow.