Nonlocal Symmetries of Nonlinear Integrable Models

It is well known that x-translation and t-translation invariance of (1) leads to the following symmetries: ux, ut of the KdV equation (1). In order to find more generalized symmetries, the concepts of recursion operators or strong symmetries, and hereditary symmetries were introduced by Olver and Fuchssteiner and used to find these symmetries [1, 2]. Furthermore, Galilean invariance of the KdV equation (1) leads to symmetry tux − 16 , which may be viewed as the origin of active research on the time-dependent symmetries and the corresponding Lie algebraic structures for nonlinear equations; and these time-dependent symmetries are connected with nonisospectral problems (see, e.g. [3–6]). Apart from the symmetries mentioned above, there exist so-called nonlocal symmetries expressed by spectral functions, e.g., σ = (φ)x is a symmetry of the KdV equation (1), where φ is a spectral function of Lax pair