NONLOCAL SYMMETRIES PAST, PRESENT AND FUTURE

Nonlocal symmetries entered the literature in the Eighties of the last century largely through the work of Peter Olver. It was observed that there could be gain of symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. In each case the source of the ‘new’ symmetry was a nonlocal symmetry, ie a symmetry with one or more of the coefficient functions containing an integral. A considerable number of different examples and occurrences were reported by Abraham-Shrauner and Guo in the early Nineties. The role of nonlocal symmetries in the integration, indeed integrability, of differential equationswas excellently illustrated by Abraham-Shrauner, Govinder and Leachwith the equation yy00 y02 + f0(x)yp+2 + pf(x)y0yp+1 = 0 which had been touted as a trivially integrable equation devoid of any point symmetry. Further theoretical contributions were made by Govinder, Feix, Bouquet, Geronimi and others in the second half of the Nineties. This included their role in reduction of order using the nonnormal subgroup. The importance of nonlocal symmetries was enhanced by the work of Krause on the Complete Symmetry Group of the Kepler Problem. Krause’s work was furthered by Nucci and there has been considerable development of the use of nonlocal symmetries by Nucci, Andriopoulos, Cotsakis and Leach. The determination of the Complete Symmetry Group for integrable systems such as the simplest version of the Ermakov equation, y00 = y 3, which possesses the algebra sl(2,R) has proven to be highly nontrivial and requires some nonintuitive nonlocal symmetries. The determination of the nonlocal symmetries required to specify completely the differential equations of nonintegrableand/or chaotic systems remains largely an open question.