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.