On travelling wave solutions of a generalized Davey–Stewartson system

The generalized Davey–Stewartson (GDS) equations, as derived by Babaoglu & Erbay (2004, Int. J. Non-Linear Mech., 39, 941–949), is a system of three coupled equations in (2 + 1) dimensions modelling wave propagation in an infinite elastic medium. The physical parameters (γ, m1, m2, λ and n) of the system allow one to classify the equations as elliptic–elliptic–elliptic (EEE), elliptic–elliptic–hyperbolic (EEH), elliptic–hyperbolic–hyperbolic (EHH), hyperbolic–elliptic–elliptic (HEE), hyperbolic–hyperbolic–hyperbolic (HHH) and hyperbolic–elliptic–hyperbolic (HEH) (Babaoglu et al., 2004, preprint). In this note, we only consider the EEE and HEE cases and seek travelling wave solutions to GDS systems. By deriving Pohozaev-type identities we establish some necessary conditions on the parameters for the existence of travelling waves, when solutions satisfy some integrability conditions. Using the explicit solutions given in Babaoglu & Erbay (2004) we also show that the parameter constraints must be weaker in the absence of such integrability conditions.