Decay of solutions to one-dimensional inhomogeneous nonlinear Schrödinger equations

TL;DR

This work develops a one-dimensional decay theory for inhomogeneous nonlinear Schrödinger equations by leveraging localized Virial–Morawetz identities with carefully chosen weights to overcome the limitations of 1D Morawetz estimates. It establishes subsequential decay in $L^2$ and $L^\infty$ on bounded spatial regions for global $H^1$-solutions without potential, and strengthens these results for odd data; it also extends the framework to include external potentials, using spectral insights from Simon to control bound-state contributions. The results cover various inhomogeneities $K(x)$ and nonlinear exponents $\sigma$, and accommodate inverse-power and Yukawa-type potentials, providing decay criteria with potential implications for energy scattering in 1D INLS. Altogether, the paper advances understanding of long-time dynamics and scattering in 1D INLS and offers techniques potentially adaptable to broader settings where classical Morawetz methods fail in low dimensions.

Abstract

We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schrödinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension remain less explored due to the absence of effective Morawetz inequalities. For equations without external potential, by establishing a localized Virial-Morawetz identity, we derive decay estimates in the context of the $L^r$-norm for global solutions within a compact domain as a time subsequence approaches infinity. This decay result can be applied to obtain a criterion for energy scattering. Additionally, by establishing another type of Virial-Morawetz identity under more strict conditions, we demonstrate the decay result for odd solutions for any time sequence that approaches infinity. Utilizing some results about bound states proved by Barry Simon, we also show that similar decay results hold for the global odd solutions of equations with suitable external potentials that contain inverse power type and Yukawa-type potentials.

Theorems & Definitions (39)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• remark 1
• Theorem 1.4
• remark 2
• Lemma 2.1: Simon1976, Klaus1977
• remark 3
• Lemma 2.2: Gagliardo-Nirenberg inequality
• Lemma 2.3