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Additional symmetries for integrable equations and conformal algebra representation. (English) Zbl 0618.35107
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.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs
35G20 Nonlinear higher-order PDEs
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