论文标题
全球适应性和二维立方聚焦非线性Schrödinger系统的散射
Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system
论文作者
论文摘要
在本文中，我们证明了在$ \ mathbb {r}^2 $上以$ l_x^2H^1（\ MathBb {r}^2 \ times^2 \ times times \ Mathbb {Z} $ \ Mathbb {r_x^2H^1（\ l_x^2H^1）低于阈值的全球焦点无限耦合的非线性耦合的非线性schrödinger系统。我们首先建立了基态的变分表征，并得出了全球良好的和散射的阈值。然后，我们通过浓度 - 紧凑型/刚度法显示了全球良好的和散射的距离，其中通过在B. Dodson的质量关键非线性非线性Schrödinger方程的证明中适应该论点来排除几乎周期性的解决方案。作为立方聚焦无限耦合的非线性Schödinger系统散射的副产品，我们在小圆柱体上获得了立方焦点非线性schrödinger方程的散射，这是焦点非线性schrödingerschrödingerschrödinger方程的第一个大数据散射结果。在文章中，我们还展示了二维$ n- $ n- $耦合的全局良好性和散射，以$ \ weft（l^2（\ mathbb {r}^2）\ right）\ right）^n $中的$ n- $ n- copepic focus focuscic focupting focus focupting focupting focupling focupling cutic nonelearschrödinger系统^n $。
In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schrödinger system on $\mathbb{R}^2$ below the threshold in $L_x^2h^1(\mathbb{R}^2\times \mathbb{Z})$. 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 $\left(L^2(\mathbb{R}^2) \right)^N$.
