On the existence of special solutions of the generalized Davey–Stewartson system

In this note we show that for certain choice of parameters the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system admits time-dependent travelling wave solutions of the kind given in [V.A. Arkadiev, A.K. Pogrebkov, M.C. Polivanov, Inverse scattering transform method and soliton solutions for Davey–Stewartson II equation, Physica D 36 (1989) 189–197] for the hyperbolic Davey–Stewartson system. These solutions lead to radial solutions as well. We also find the sufficient conditions for non-existence of travelling wave solutions for the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system by using the point of view developed in [A. Eden, T.B. Gürel, E. Kuz, Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system, IMA J. Appl. Math. (in press)].