Four Symmetries of the KdV Equation

Alexander G. Rasin, Jeremy Schiff

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Article number68
JournalJournal of Nonlinear Science
Issue number5
StatePublished - Oct 2022

Bibliographical note

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


  • Commutator algebra
  • Generating symmetry
  • KdV equation
  • Nonlocal symmetry
  • Symmetry


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