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On a nonlocal non-linear inverse scattering problem

Saumyajit Das, Susovan Pramanik

Abstract

In this article, we study the inverse scattering problem for the nonlinear fractional Helmholtz equation with cubic nonlinearity in three dimensions, where we recover a compactly supported potential from scattering amplitude.

On a nonlocal non-linear inverse scattering problem

Abstract

In this article, we study the inverse scattering problem for the nonlinear fractional Helmholtz equation with cubic nonlinearity in three dimensions, where we recover a compactly supported potential from scattering amplitude.
Paper Structure (5 sections, 12 theorems, 83 equations)

This paper contains 5 sections, 12 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Resolvent Estimate and Limiting Absorption Principle
  3. Fundamental Solution
  4. Scattered Field and Scattering Amplitude
  5. Inverse Problem

Key Result

Theorem 1.1

Let $u$ satisfies where $k>0$ and $Q(x)\in \mathrm{L}^{\infty}({\mathbb R}^d)$ is compactly supported. Let the initial wave function is given by where $\| g\|_{\mathrm{L}^{2}({\mathbb R}^d)}\le \epsilon$ for some $\epsilon>0$. Furthermore, assume Then, there exists a unique solution '$u$' to the equation Frac Helmholtz eq no index, satisfying provided $u$ satisfy the following Sommerfield radi

Theorems & Definitions (18)

  • Theorem 1.1: shen2025complex
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2: shen2025complex
  • Proposition 2.3: das2025inverseshen2025complex
  • Lemma 2.4
  • Proposition 2.5
  • ...and 8 more