Almost cubic nonlinear schrödinger equation: existence, uniqueness and scattering

Abstract

In this thesis, a uni ed treatment is given for a class of nonlinear non-local 2D elliptic and hyperbolic Schrödinger equation which includes the 2D nonlinear Schrödinger (NLS) equation with a purely cubic nonlinearity, Davey-Stewartson (DS) system in the hyperbolic-elliptic (HE) and elliptic-elliptic (EE) cases and the generalized Davey- Stewartson (GDS) system in the hyperbolic-elliptic-elliptic (HEE) and elliptic-ellipticelliptic (EEE) cases. Local in time existence and uniqueness of solutions are established for the Cauchy problem when initial data is in L2(R2), H1(R2), H2(R2) and in = H1(R2) \ L2(jxj2 dx) and the maximal time of existence for the solutions all agree. Conserved quantities corresponding to mass, momentum, energy are derived, as well as scale and pseudo-conformal invariance of solutions. Virial identity is also established and its relation to pseudo-conformal invariance is discussed. Various routes to global existence of solutions are also explored in the elliptic case, namely, for small mass solutions in L2(R2); in the defocusing case for solutions in H1(R2) and nally in the focusing case for H1(R2)-solutions with subminimal mass. In all such cases the scattering of such solutions in L2(R2) and topologies are discussed. Moreover, in the focusing case when initial energy is negative, it is shown that solutions in blow-up. The existence and uniqueness results are also considered for more general nonlinearities.