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

TL;DR

This paper investigates the long-time behavior of the mass-critical nonlinear Schrödinger equation on the real line, establishing finite-dimensional approximations, non-squeezing properties, and homogenization results for inhomogeneous models.

Contribution

It introduces a finite-dimensional Hamiltonian system approximation for the NLS and analyzes the non-squeezing property and homogenization in inhomogeneous cases.

Findings

Solution can be approximated by a finite-dimensional Hamiltonian system.

Established the non-squeezing property for the truncated NLS on the torus.

Proved convergence of inhomogeneous NLS solutions to the homogeneous model as the inhomogeneity parameter grows.

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.