On the Calogero–Degasperis–Fokas equation in (2+1) dimensions

In this paper we study a (2+1)-dimensional integrable Calogero–Degasperis–Fokas equation derivable by using a method proposed by Calogero. A catalogue of classical symmetry reductions are given. These reductions to partial differential equations in (1+1) admit symmetries which lead to further reductions, i.e., to second-order ordinary differential equations. These ODEs provide several classes of solutions; all of them are expressible in terms of known functions, some of them are expressible in terms of the second and third Painleve trascendents. The corresponding solutions of the (2+1)-dimensional equation, involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviour. Indeed by making appropriate choices for the arbitrary functions, we exhibit solitary waves, coherent structures and bound states.