An Effective Algorithm for Finding Multidimensional Conservation Laws for Integrable Systems of Hydrodynamic Type

Makridin, Zakhar V.

doi:10.1134/s0040577918020071

Cited by 3 publications

(4 citation statements)

References 12 publications

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“…At the first stage of our investigation (see details in Ref. ), we were able to prove existence of infinitely many such quasilocal conservation laws for the dispersionless limit of the Kadomtsev‐Petviashvili equation, reducing corresponding computations to infinitely many overdetermined systems in involutions, whose general solutions depend on arbitrary constants only. Instead of infinitely many particular calculations, in this paper we solved just one such an over‐determined system, whose general solution determines a generating function of three‐dimensional quasilocal conservation laws.…”

Section: Resultsmentioning

confidence: 99%

“…The Benney hydrodynamic chain and its first higher commuting flow together possess infinitely many three‐dimensional local conservation laws

\begin{matrix} true \\ right {(A_{k, m} (H_{0}, H_{1}, \dots, H_{k}))}_{y} + {(B_{k, m} (H_{0}, H_{1}, \dots, H_{k}, H_{k + 1}))}_{t} + {(C_{k, m} (H_{0}, H_{1}, \dots, H_{k}, H_{k + 1}))}_{x} = 0, \end{matrix}

where

A_{k, m}, B_{k, m}, C_{k, m}

are polynomials with respect to

H_{i}

, and the index

k = 0, 1, 2, \dots

, whereas the index

m = 0, 1, \dots, k

. For instance, the first 10 three‐dimensional conservation laws have densities

A_{k, m}

and the corresponding fluxes

B_{k, m}, C_{k, m}

are given in the Table below.…”

Section: The Benney Hydrodynamic Hierarchymentioning

confidence: 99%

“…The Benney hydrodynamic chain (4) and its first higher commuting flow (7) together possess infinitely many three-dimensional local conservation laws 19 (…”

Section: The Benney Hydrodynamic Hierarchymentioning

confidence: 99%

“…Then one can choose two first equations ( 1 ) = ( 2 − 2 1 ) , ( 2 ) = ( 3 − 1 2 ) of the first hydrodynamic chain and the first equation ( 1 ) = ( 3 − 2 1 2 + 3 1 ) of the second hydrodynamic chain. Eliminating 3 , finally we obtain Mikhalë v system (19). We will work in the -variables.…”

Section: The Mikhalëv Hydrodynamic Hierarchymentioning

confidence: 99%

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Multi‐dimensional conservation laws and integrable systems

Makridin

Pavlov

2019

Stud Appl Math

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In this paper, we introduce a new property of two‐dimensional integrable hydrodynamic chains—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Infinitely many local three‐dimensional conservation laws for the Benney commuting hydrodynamic chains are constructed. As a by‐product, we established a new method for computation of local conservation laws for three‐dimensional integrable systems. The Mikhalëv equation and the dispersionless limit of the Kadomtsev‐Petviashvili equation are investigated. All known local and infinitely many new quasilocal three‐dimensional conservation laws are presented. Also four‐dimensional conservation laws are considered for couples of three‐dimensional integrable quasilinear systems and for triplets of corresponding hydrodynamic chains.

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Section: Resultsmentioning

confidence: 99%

“…The Benney hydrodynamic chain and its first higher commuting flow together possess infinitely many three‐dimensional local conservation laws

\begin{matrix} true \\ right {(A_{k, m} (H_{0}, H_{1}, \dots, H_{k}))}_{y} + {(B_{k, m} (H_{0}, H_{1}, \dots, H_{k}, H_{k + 1}))}_{t} + {(C_{k, m} (H_{0}, H_{1}, \dots, H_{k}, H_{k + 1}))}_{x} = 0, \end{matrix}

where

A_{k, m}, B_{k, m}, C_{k, m}

are polynomials with respect to

H_{i}

, and the index

k = 0, 1, 2, \dots

, whereas the index

m = 0, 1, \dots, k

. For instance, the first 10 three‐dimensional conservation laws have densities

A_{k, m}

and the corresponding fluxes

B_{k, m}, C_{k, m}

are given in the Table below.…”

Section: The Benney Hydrodynamic Hierarchymentioning

confidence: 99%

“…The Benney hydrodynamic chain (4) and its first higher commuting flow (7) together possess infinitely many three-dimensional local conservation laws 19 (…”

Section: The Benney Hydrodynamic Hierarchymentioning

confidence: 99%

Section: The Mikhalëv Hydrodynamic Hierarchymentioning

confidence: 99%

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Multi‐dimensional conservation laws and integrable systems

Makridin

Pavlov

2019

Stud Appl Math

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

Krasil’shchik

Sergyeyev

2019

Ann. Henri Poincaré

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Multidimensional conservation laws and integrable systems II

Makridin

Павлов

2021

Stud Appl Math

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In this paper we continue investigation of a new property of two-dimensional integrable systems-existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional commuting flows. Multicomponent two-dimensional hydrodynamic reductions of the Mikhalëv equation are considered. Infinitely many three-dimensional local conservation laws for the Korteweg-de Vries pair of commuting flows are constructed. Thus, we show that pairs of commuting dispersive two-dimensional systems also possess infinitely many local three-dimensional conservation laws. They can be used for averaging of multiparametric families of solutions to the Mikhalëv equation.

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An Effective Algorithm for Finding Multidimensional Conservation Laws for Integrable Systems of Hydrodynamic Type

Cited by 3 publications

References 12 publications

Multi‐dimensional conservation laws and integrable systems

Multi‐dimensional conservation laws and integrable systems

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

Multidimensional conservation laws and integrable systems II

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