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An inverse Problem for the cubic $α$-NLS in Sobolev spaces

Zachary Lee, Nataša Pavlović

TL;DR

The paper tackles the inverse problem for the defocusing cubic NLS in dimensions $d\in\{1,2,3\}$ with a space-dependent nonlinearity coefficient $\alpha(x)$. It develops an approximate-solution framework in Sobolev spaces, producing a large-amplitude, high-frequency ansatz whose leading nonlinear transport dynamics encode $\alpha$; a stability analysis shows the actual solution remains close to the approximation. By evaluating the true solution at carefully chosen space-time points, the authors derive a quantity $X\alpha(x_0,\xi)$ related to the X-ray transform, enabling explicit reconstruction of $\alpha$ via inverse X-ray methods in $d=2,3$ (and a 1D special case). The work provides, for the first time with approximate solutions in Sobolev spaces, explicit recovery of $\alpha$ from the NLS solution, establishing a concrete link between nonlinear wave dynamics and tomographic-type data.

Abstract

In this work, we address an inverse problem for a defocusing cubic nonlinear Schrödinger (NLS) equation in dimensions $d\in\{1, 2,3\}$ in a range of Sobolev spaces $H^s(\mathbb{R}^d)$ by employing the method of approximate solutions. We recover a smooth, space-dependent and compactly supported function $α$ that controls the nonlinearity (and thus self-interaction strength) in a multiplicative fashion. To the best of our knowledge, this is the first work based on approximate solutions in Sobolev spaces that treats an inverse problem for the NLS and provides explicit recovery of $α$.

An inverse Problem for the cubic $α$-NLS in Sobolev spaces

TL;DR

The paper tackles the inverse problem for the defocusing cubic NLS in dimensions with a space-dependent nonlinearity coefficient . It develops an approximate-solution framework in Sobolev spaces, producing a large-amplitude, high-frequency ansatz whose leading nonlinear transport dynamics encode ; a stability analysis shows the actual solution remains close to the approximation. By evaluating the true solution at carefully chosen space-time points, the authors derive a quantity related to the X-ray transform, enabling explicit reconstruction of via inverse X-ray methods in (and a 1D special case). The work provides, for the first time with approximate solutions in Sobolev spaces, explicit recovery of from the NLS solution, establishing a concrete link between nonlinear wave dynamics and tomographic-type data.

Abstract

In this work, we address an inverse problem for a defocusing cubic nonlinear Schrödinger (NLS) equation in dimensions in a range of Sobolev spaces by employing the method of approximate solutions. We recover a smooth, space-dependent and compactly supported function that controls the nonlinearity (and thus self-interaction strength) in a multiplicative fashion. To the best of our knowledge, this is the first work based on approximate solutions in Sobolev spaces that treats an inverse problem for the NLS and provides explicit recovery of .

Paper Structure

This paper contains 13 sections, 8 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation
  4. Preliminary Results
  5. Construction of Approximate Solution
  6. The leading order equation for $a_0$
  7. Higher order terms
  8. Proof of Theorem \ref{['Main Theorem']}
  9. Recovery
  10. Proof of Theorem \ref{['Recovery Theorem']}
  11. Definitions
  12. Proof of Proposition 2.3 and 2.4
  13. Proof of Proposition 2.6

Key Result

Theorem 1.1

Let $d=1,2,3$, $\alpha\in C_c^\infty({\mathbb{R}}^d)$, $\alpha \ge 0 ,T_0>0$ such that $\operatorname{supp}\alpha \subseteq B(0,T_0)$ and $T>\frac{T_0}{2}$. Also let $\xi \in {\mathbb{R}}^d$ such that $|\xi|=1$ and let $h$ be such that $0<h<1$. Suppose $\psi\in C_c^\infty ({\mathbb{R}}^d)$ and $\ope If $u\in C_t^0 H^{d/2+\delta}_x([-Th, Th]\times {\mathbb{R}}^d)$, with $\delta(d)>0$ for $d\in\{1,2

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4: Stability
  • Lemma 3.1
  • ...and 9 more