THE GAMBIER MAPPING, REVISITED

We examine critically the Gambler equation and show that it is the gen eric linearisable equation containing, as reductions, all the second-o rder equations which are integrable through linearisation, We then int roduce the general discrete form of this equation, the Gambler mapping , and present conditions for its integrability. Finally, we obtain the reductions of the Gambler mapping, identify their integrable forms an d compute their continuous limits, (C) 1998 Elsevier Science B.V. All rights reserved.