Stability of nonlinear recovery from scattering and modified scattering maps

TL;DR

The paper addresses the inverse problem of recovering a localized inhomogeneity $a(x)$ in a 1D nonlinear Schrödinger equation from scattering data. It develops a quantitative framework using Born approximations and approximate identities, handling both the standard scattering map and the modified scattering map, and distinguishes the regimes $p>2$ and $p=2$. The main results are Hölder-type stability for $p\in(2,4]$ and logarithmic stability at $p=2$ in the standard scattering problem, and logarithmic stability for the modified scattering map; a new $p=2$ approximate-identity lemma (and its use in the modified map) is a key technical contribution. These results provide explicit stability estimates with constants depending on norms of $a$ and demonstrate identifiability of $a$ from nonlinear scattering data, with implications for inverse problems in nonlinear dispersive PDEs.

Abstract

We prove stability estimates for the recovery of the nonlinearity from the scattering or modified scattering map for one-dimensional nonlinear Schrödinger equations. We consider nonlinearities of the form $a(x) |u|^p u$ for $p\in [2,4]$ and $[1+a(x)]|u|^2 u$, where $a$ is a localized function. In the first case, we show that for $p\in(2,4]$ we may obtain a Hölder-type stability estimate for recovery via the scattering map, while for $p=2$ we obtain a logarithmic stability estimate. In the second case, we show a logarithmic stability estimate for recovery via the modified scattering map.

Theorems & Definitions (14)

• Theorem 1.1: Small-data scattering
• Theorem 1.2: Stability for \ref{['nls1']}, $p>2$
• Theorem 1.3: Stability for \ref{['nls1']}, $p=2$
• Theorem 1.4: Small-data modified scattering, ChenMurphy1
• Theorem 1.5: Stability for \ref{['nls2']}
• proof : Proof of Theorem \ref{['T:scatter1']}
• Lemma 2.1: Born approximation
• Lemma 2.2: Structure of the modified scattering map
• proof
• Lemma 3.1: Approximate identity, $p>2$