Dispersion chain of Vlasov equations

Abstract On the basis of the Vlasov chain of equations, a new infinite dispersion chain of equations is obtained for the distribution functions of mixed higher order kinematical values. In contrast to the Vlasov chain, the dispersion chain contains distribution functions with an arbitrary set of kinematical values and has a tensor form of writing. For the dispersion chain, new equations for mixed Boltzmann functions and the corresponding chain of conservation laws for fluid dynamics are obtained. The probability is proved to be a constant value for a particle to belong the region where the quasi-probability density is negative (Wigner function).

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On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽

10.3390/sym13050878 ◽

2021 ◽

Vol 13 (5) ◽

pp. 878

Author(s):

Alexei Cheviakov ◽

Denys Dutykh ◽

Aidar Assylbekuly

Keyword(s):

Wave Equation ◽

Conservation Laws ◽

Numerical Simulations ◽

Solitary Waves ◽

Local Symmetry ◽

Symmetry And Conservation Laws ◽

Higher Order ◽

Special Equation ◽

Long Wave ◽

Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

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Hyperbolic three-string vertex

Journal of High Energy Physics ◽

10.1007/jhep08(2021)035 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Atakan Hilmi Fırat

Keyword(s):

Boundary Value Problem ◽

Conservation Laws ◽

Riemann Surfaces ◽

Higher Order ◽

Local Coordinates ◽

Pants Decomposition ◽

Fuchsian Equation ◽

String Amplitudes ◽

Hyperbolic Riemann Surfaces ◽

Liouville’S Equation

Abstract We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

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Canonical global symmetry and quantal conservation laws for a system with singular higher order Lagrangian

International Journal of Theoretical Physics ◽

10.1007/bf02435741 ◽

1997 ◽

Vol 36 (2) ◽

pp. 431-439

Author(s):

Zi-ping Li ◽

Li Wang

Keyword(s):

Conservation Laws ◽

Higher Order ◽

Global Symmetry ◽

Higher Order Lagrangian

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Multisymplectic formulation of fluid dynamics using the inverse map

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.1892 ◽

2007 ◽

Vol 463 (2086) ◽

pp. 2671-2687 ◽

Cited By ~ 35

Author(s):

C.J Cotter ◽

D.D Holm ◽

P.E Hydon

Keyword(s):

Fluid Dynamics ◽

Conservation Laws ◽

Fluid Particle ◽

Hamilton’S Principle ◽

Hamilton's Principle ◽

Numerical Integrators ◽

Fluid Particles ◽

Infinite Set ◽

Circulation Theorem

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.

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