Two-dimensional nonlinear Schro¨dinger equations and their properties

Abstract This paper studies a general class of two-dimensional systems of the cubic nonlinear Schrodinger type (2DNLS), defined by i∂ t q + O 1 q = pq and O 2 p = O 3 (q ∗ q) , where each O n  D ( n ) ij ∂ i ∂ 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 O n . We obtain all integrable cases, including the known cases Davey-Stewartson I, II and II′, as well as other known integrable cases. The 2DNLS is modulationally stable if D (1) D (2) D (3) > 0↭k , where D (n) = k i k j D (n) ij . 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 (n) ij : O 1 and O 2 must both be elliptic, and for that choice of variables for which D (1) ij and D (2) ij both have positive signature, D (3) ij 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.