Asymptotic behavior of mass-critical Schr\"odinger equation in $ \mathbb{R}$

Abstract

In this paper, we study the long-time behavior for the mass-critical nonlinear Schr\"odinger equation on the line \[ i\partial_t u + \partial_x^2 u = |u|^4 u, u(0, x) = u_0 \in L_x^2(\Bbb R). \] The global well-posedness and scattering for this equation was solved in Dodson [Amer. J. Math. (2016)]. Inspired by the pioneering work of Killip-Visan-Zhang [Amer. J. Math. (2021)], we show that solution can be approximated by a finite-dimensional Hamiltonian system. This system is the nonlinear Schr\"odinger equation on the rescaled torus $\Bbb R/(L_n\Bbb Z)$ with Fourier truncated nonlinear term. To prove this, we introduce the Fourier truncated mass-critical NLS on $\mathbb{R}$. First, we establish the uniformly global space-time bound for this truncated model on $\mathbb{R}$. Second, we show that the truncated NLS on rescaled torus can be approximated by the truncated equation on $\Bbb R$. Then, using the Gromov theorem, we can show the non-squeezing property for the truncated NLS on torus. The last step to show the non-squeezing property for original NLS is to connect the solution with truncated nonlinearity and a single equation in $\mathbb{R}$, which can be done by performing the nonlinear profile decomposition. Our second result is to study the homogenization of the mass-critical inhomogeneous NLS, where we add a $L^\infty$ function $h(nx)$ in front of the nonlinear term. Based on the method of Ntekoume [Comm. PDE, (2020)], we give the sufficient condition on $h$ such that the scattering holds for this inhomogeneous model and show that the solution to inhomogeneous converges to the homogeneous model when $n\to\infty$. As a corollary, we can transfer the non-squeezing property from homogeneous model to inhomogeneous.