2-DIMENSIONAL NONLINEAR SCHRODINGER-EQUATIONS AND THEIR PROPERTIES

This paper studies a general class of two-dimensional systems of the c ubic nonlinear Schrodinger type (2DNLS), defined by i partial derivati ve(t)q + O1q = pq and O2p = O3(q q), where each O(n) = D(ij)(n) parti al derivative(i) partial derivative(j), n = 1,2,3, is a linear, second -order, operator with constant coefficients. This class generalizes th e Djordjevic-Redekopp (DR) system, which has previously been encounter ed in the context of water waves. Integrability is characterized simpl y in terms of covariant conditions on the O(n). We obtain all integrab le cases, including the known cases Davey-Stewartson I, II and II', as well as other known integrable cases. The 2DNLS is modulationally sta ble if D(1)D(2)D(3) > 0 for-all k, where D(n) = k(i)k(j)D(ij)(n). All other regimes are modulationally unstable and have projections satisfy ing the ordinary (1D) NLS with soliton solutions, though in all known cases these 1D solitons are unstable with respect to transverse pertur bations. The self-focusing regime is characterized by the eigenvalues of the D(ij)(n): O1 and O2 must both be elliptic, and for that choice of variables for which D(ij)(1) and D(ij)(2) both have positive signat ure, D(ij)(3) must have at least one negative eigenvalue. The self-foc using regime is distinct from the modulationally stable regime and als o from the integrable regime, while the integrable cases may be modula tionally stable or unstable. There are no soliton solutions known in t hose integrable cases that are modulationally stable, whereas those in tegrable cases in which 2D solitons are known correspond to the modula tional instability regime.