Two-Dimensional Nonlinear Schrodinger Equations and Their Properties

This paper studies a general class of two-dimensional systems of the cubic nonlinear Schrodinger type 2DNLS, defined by i del sub t of q O1q pq and O2p O3qq, where each O sub n is equated to D sub ij superscript n del sub i del sub j, n 1,2,3, is a linear, second-order, operator with constant coefficients. This class generalizes the Djordjevic-Redekopp DR system, which has previously been encountered in the context of water waves. Integrability is characterized simply in terms of covariant conditions on the On. We obtain all integrable cases, including the known cases Davey-Stewartson I and II as well as other known integrable cases. All other regimes are modulationally unstable and have projections satisfying the ordinary 1D NLS with soliton solutions, though in all known cases these 1D solitons are unstable with respect to transverse perturbations. The self-focusing regime is characterized by the eigenvalues of the D sub ij superscript n O1 and O2 must both be elliptic, and for that choice of variables for which D sub ij superscript 1 and D sub ij superscript 2 both have positive signature, D sub ij superscript 3 must have at least one negative eigenvalue. The self-focusing regime is distinct from the modulationally stable regime and also from the integrable regime, while the integrable cases may be modulationally stable or unstable. There are no soliton solutions known in those integrable cases that are modulationally stable, whereas those integrable cases in which 2D solitons are known correspond to the modulational instability regime.