Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

Abstract

In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schrödinger system on R 2 below the threshold in L 2x h 1 (R 2 × Z). We first establish the variational characterization of the ground state, and derive the threshold of the global well-posedness and scattering. Then we show the global well-posedness and scattering below the threshold by the concentration-compactness/rigidity method, where the almost periodic solution is excluded by adapting the argument in the proof of the mass-critical nonlinear Schrödinger equations by B. Dodson. As a byproduct of the scattering of the cubic focusing infinite coupled nonlinear Schödinger system, we obtain the scattering of the cubic focusing nonlinear Schrödinger equation on the small cylinder, this is the first large data scattering result of the focusing nonlinear Schrödinger equations on the cylinders. In the article, we also show the global well-posedness and scattering of the two dimensional N −coupled focusing cubic nonlinear Schrödinger system in L 2 (R 2 )N .

Small data finite-time blow-up solutions for the nonlinear Schrödinger equation in general dimensions

“…In the focusing case, Chen-Guo-Hwang-Yoon [9] proved the global well-posedness and scattering of the large scale solution to (1.16) on R 2 × T by proving the global wellposedness and scattering of the resonant Schrödinger system below the ground state.…”

On scattering for generalized NLS on waveguide manifolds

“…Threshold assumptions are necessary and new ingredients are needed to handle this type of problems. See [45] for a recent global well-posedness result, see [12,13,25,27] for the Euclidean results and see [6,31] for some very recent scattering result. 4.…”

On scattering asymptotics for the 2D cubic resonant system

“…Threshold assumptions are necessary and new ingredients are needed to handle this type of problems. See [37] for a global well-posedness result and [7,28] for two very recent scattering result. Moreover, see [14,24,27] for the Euclidean result.…”