[Paper Review] Long-time dynamics of small solutions to 1$d$ cubic nonlinear Schrödinger equations with a trapping potential

Chen Gong|arXiv (Cornell University)|Jun 18, 2021

Advanced Mathematical Physics Problems63 references4 citations

TL;DR

This paper establishes the long-time dynamics of small solutions to the 1D cubic nonlinear Schrödinger equation with a trapping potential, showing that solutions decompose into a solitary wave and a radiation term with modified scattering. Using distorted Fourier analysis and dispersive estimates, it proves global existence and precise asymptotic behavior, including logarithmic phase corrections and soliton persistence due to the potential's negative eigenvalue.

ABSTRACT

In this paper, we analyze the long-time dynamics of small solutions to the $1d$ cubic nonlinear Schrödinger equation (NLS) with a trapping potential. We show that every small solution will decompose into a small solitary wave and a radiation term which exhibits the modified scattering. In particular, this result implies the asymptotic stability of small solitary waves. Our analysis also establishes the long-time behavior of solutions to a perturbation of the integrable cubic NLS with the appearance of solitons.

Motivation & Objective

• To understand the long-time behavior of small-norm solutions to the 1D cubic NLS with a trapping potential that has one negative eigenvalue.
• To extend modified scattering theory to non-integrable systems with discrete spectrum by incorporating space-time resonance and dispersive analysis.
• To describe the asymptotic decomposition of solutions into a solitary wave and a radiation term under slow dispersion and critical scattering.
• To analyze perturbations of the integrable cubic NLS where solitons emerge due to the potential's spectral structure.
• To establish global-in-time bounds and refined estimates for solutions using a priori analysis in weighted function spaces.

Proposed method

• Uses distorted Fourier transforms to handle the linear Schrödinger operator with a potential, particularly its discrete spectrum.
• Applies space-time resonance analysis to decompose the nonlinear interaction into resonant and non-resonant parts.
• Employs dispersive estimates and weighted $L^1$ conditions on the potential to control nonlinear interactions.
• Implements a fixed-point argument in a tailored function space $X$ combining $L^ ho_t H^1_x$, $L^4_t L^rown_x$, and time-weighted norms to ensure global existence.
• Performs a priori estimates and bootstrap arguments in the space $Y_T$ to control growth and ensure convergence.
• Decomposes the solution into low- and high-frequency components in time to refine estimates on the radiation term.

Experimental results

Research questions

• RQ1 How do small solutions to the 1D cubic NLS with a trapping potential behave as $t \to \infty$?
• RQ2 Can modified scattering be established when the linear operator has a negative eigenvalue, leading to solitary wave formation?
• RQ3 What is the precise asymptotic structure of the solution when the potential introduces a discrete eigenvalue and the nonlinearity is critical?
• RQ4 How does the presence of a solitary wave affect the long-time decay and phase correction of the radiation component?
• RQ5 To what extent can dispersive and spectral methods be combined to analyze non-integrable NLS with potential and discrete spectrum?

Key findings

• Every small solution to the 1D cubic NLS with a trapping potential decomposes into a solitary wave and a radiation term that exhibits modified scattering with logarithmic phase corrections.
• The solution remains globally bounded in $H^1$ and the $H^1$ norm is uniformly controlled via conservation laws and smallness assumptions.
• The radiation component decays like $|t|^{-1/2}$ in $L^rown_x$, with a logarithmic phase correction due to the nonlinearity.
• The solitary wave persists asymptotically due to the negative eigenvalue of the linear operator $-\partial_{xx} + V$, which supports a bound state.
• The asymptotic behavior is described via a refined function space $X$ with time-weighted norms, ensuring global existence and sharp estimates.
• The analysis extends to perturbations of the integrable cubic NLS, showing that solitons emerge and the solution remains close to the integrable structure under small nonlinear perturbations.