Four Symmetries of the KdV equation
Abstract
We identify 4 nonlocal symmetries of KdV depending on a parameter. 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, 3 of the 4 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 3 symmetries that commute in the first version of the algebra can all be regarded as infinitesimal double Bäcklund transformations.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.01550
 Bibcode:
 2021arXiv210801550R
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 58D19 17B80
 EPrint:
 20 pages