Abstract
We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified ‘dressing’ method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S 1 algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators \(\lambda ^m \partial _\lambda \), with integer m, p. Some additional results about Kac-Moody algebra applications are presented.
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Orlov, A.Y., Schulman, E.I. Additional symmetries for integrable equations and conformal algebra representation. Lett Math Phys 12, 171–179 (1986). https://doi.org/10.1007/BF00416506
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DOI: https://doi.org/10.1007/BF00416506