Nonlocal symmetry and similarity reductions for the Drinfeld-Sokolov-Satsuma-Hirota system

The nonlocal symmetry of the Drinfeld-Sokolov-Satsuma-Hirota system is obtained from the known Lax pair, and infinitely many nonlocal symmetries are given by introducing the internal parameters. Then the nonlocal symmetry is localized to a prolonged system by introducing suitable auxiliary dependent variables. By applying the classical Lie symmetry method to this prolonged system, two main results are obtained: a new type of finite symmetry transformation is derived, which can generate new solutions from old ones; some exact interaction solutions among solitons and other complicated waves including periodic cnoidal wave and Painlevé waves are derived through similarity reductions.