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Asymptotic formulas for phase recovering from phaseless data of biharmonic waves at a fixed frequency

Yuxiang Cheng, Xiaoxu Xu, Haiwen Zhang

Abstract

This paper focuses on phase retrieval from phaseless total-field data in biharmonic scattering problems. We prove that a phased biharmonic wave can be uniquely determined by the modulus of the total biharmonic wave within a nonempty domain. As a direct corollary, the uniqueness for the inverse biharmonic scattering problem with phaseless total-field data is established. Moreover, using the Atkinson-type asymptotic expansion, we derive explicit asymptotic formulas for the problem of phase retrieval.

Asymptotic formulas for phase recovering from phaseless data of biharmonic waves at a fixed frequency

Abstract

This paper focuses on phase retrieval from phaseless total-field data in biharmonic scattering problems. We prove that a phased biharmonic wave can be uniquely determined by the modulus of the total biharmonic wave within a nonempty domain. As a direct corollary, the uniqueness for the inverse biharmonic scattering problem with phaseless total-field data is established. Moreover, using the Atkinson-type asymptotic expansion, we derive explicit asymptotic formulas for the problem of phase retrieval.
Paper Structure (7 sections, 18 theorems, 96 equations)

This paper contains 7 sections, 18 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Uniqueness of phase retrieval
  4. Explicit asymptotic formulas for phase retrieval
  5. Three-dimensional case
  6. Two-dimensional case
  7. Conclusion

Key Result

Lemma 2.1

Let $(u_H,u_M)$ solve 1'--3". Assume that both $u_H$ and $u_M$ possess normal derivatives on the boundary. For $x\in{\mathbb R}^m\backslash\overline{D}$ we have where $\Phi_k(x,y)$ and $\Phi_{ik}(x,y)$ are fundamental solutions of Helmholtz equation 1' and modified Helmholtz equation 1", respectively.

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 26 more