TY - JOUR

T1 - Four Symmetries of the KdV Equation

AU - Rasin, Alexander G.

AU - Schiff, Jeremy

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/10

Y1 - 2022/10

N2 - We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify four nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these are nonlocal symmetries, their commutator algebra is not uniquely determined, and we present three possibilities for the algebra. In the first version, three of the four symmetries commute; this shows that it is possible to add further (nonlocal) commuting flows to the standard KdV hierarchy. The second version of the commutator algebra is consistent with Laurent expansions of the symmetries, giving rise to an infinite-dimensional algebra of hidden symmetries of KdV. The third version is consistent with asymptotic expansions for large values of the parameter, giving rise to the standard commuting symmetries of KdV, the infinite hierarchy of “additional symmetries,” and their traditionally accepted commutator algebra (though this also suffers from some ambiguity as the additional symmetries are nonlocal). We explain how the three symmetries that commute in the first version of the algebra can all be regarded as infinitesimal double Bäcklund transformations. The four generating symmetries incorporate all known symmetries of the KdV equation, but also exhibit some remarkable novel structure, arising from their nonlocality. We believe this structure to be shared by other integrable PDEs.

AB - We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify four nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these are nonlocal symmetries, their commutator algebra is not uniquely determined, and we present three possibilities for the algebra. In the first version, three of the four symmetries commute; this shows that it is possible to add further (nonlocal) commuting flows to the standard KdV hierarchy. The second version of the commutator algebra is consistent with Laurent expansions of the symmetries, giving rise to an infinite-dimensional algebra of hidden symmetries of KdV. The third version is consistent with asymptotic expansions for large values of the parameter, giving rise to the standard commuting symmetries of KdV, the infinite hierarchy of “additional symmetries,” and their traditionally accepted commutator algebra (though this also suffers from some ambiguity as the additional symmetries are nonlocal). We explain how the three symmetries that commute in the first version of the algebra can all be regarded as infinitesimal double Bäcklund transformations. The four generating symmetries incorporate all known symmetries of the KdV equation, but also exhibit some remarkable novel structure, arising from their nonlocality. We believe this structure to be shared by other integrable PDEs.

KW - Commutator algebra

KW - Generating symmetry

KW - KdV equation

KW - Nonlocal symmetry

KW - Symmetry

UR - http://www.scopus.com/inward/record.url?scp=85135438024&partnerID=8YFLogxK

U2 - 10.1007/s00332-022-09831-8

DO - 10.1007/s00332-022-09831-8

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AN - SCOPUS:85135438024

SN - 0938-8974

VL - 32

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 5

M1 - 68

ER -