Integrability of Anti-Self-Dual Vacuum Einstein Equations with Nonzero Cosmological Constant: An Infinite Hierarchy of Nonlocal Conservation Laws

We solve vacuum field equations in five-dimensional gravity with cosmological constant to determine the time-dependence of the Robertson–Walker scale factor. We discuss its cosmological implications.

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Solution of the vacuum Einstein equations with non-zero cosmological constant for a stationary cylindrically symmetric spacetime

Classical and Quantum Gravity ◽

10.1088/0264-9381/10/11/022 ◽

1993 ◽

Vol 10 (11) ◽

pp. 2401-2406 ◽

Cited By ~ 30

Author(s):

N O Santos

Keyword(s):

Cosmological Constant ◽

Einstein Equations ◽

Cylindrically Symmetric ◽

Vacuum Einstein Equations ◽

Cylindrically Symmetric Spacetime ◽

Symmetric Spacetime

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A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry074 ◽

2018 ◽

Vol 40 (1) ◽

pp. 405-421 ◽

Cited By ~ 1

Author(s):

N Chatterjee ◽

U S Fjordholm

Keyword(s):

Initial Data ◽

Conservation Laws ◽

Finite Volume Method ◽

Finite Volume ◽

Numerical Examples ◽

Continuity Equations ◽

Multiple Dimensions ◽

Nonlocal Conservation Laws ◽

Volume Method ◽

Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

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

Symmetry ◽

10.3390/sym13061077 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1077

Author(s):

Yarema A. Prykarpatskyy

Keyword(s):

Conservation Laws ◽

Hamiltonian Structure ◽

Necessary Condition ◽

Type Representation ◽

Infinite Hierarchy ◽

Polynomial Coefficients ◽

Kdv Type Equations ◽

Special Case

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.

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