Two remarks on a generalized Davey-Stewartson system

Many equations can be expressed as a cubic nonlinear Schrödinger (NLS) equation with additional terms, such as the Davey-Stewartson (DS) system [1]. As it is the case for the NLS equation, the solutions of the DS system are invariant under the pseudo-conformal transformation. For the elliptic NLS, this invariance plays a key role in understanding the blow-up profile of solutions, whereas in the hyperbolic-elliptic case of DS system an explicit blow-up profile is obtained via the pseudo-conformal invariance. An analogous system has been derived in [2] to model wave propagation in a generalized elastic medium and has been called Generalized Davey-Stewartson (GDS) system. In [3], for the hyperbolic-elliptic-elliptic and elliptic-elliptic-elliptic cases the GDS system has been expressed as a NLS equation with non-local terms. We present two results on the GDS system, both following from the pseudoconformal invariance of its solutions. In the hyperbolic-elliptic-elliptic case, under some conditions on the physical parameters, we establish a blow-up profile. These conditions turn out to be necessary conditions for the existence of a special “radial” solution. In the elliptic-elliptic-elliptic case, under milder conditions, we show the L-norms of the solutions decay to zero algebraically in time for 2 < p <∞.