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Gaussian beam interactions and inverse source problems for nonlinear wave equations

Matti Lassas, Tony Liimatainen, Valter Pohjola, Teemu Tyni

TL;DR

The paper studies inverse source problems for the semilinear wave equation $$(\Box_g+ q_1)u + q_2 u^2 = F$$ on a globally hyperbolic Lorentzian manifold and proves that the coefficients $(q_1,q_2)$ and the source $F$ are recoverable from local measurements up to a natural gauge symmetry. The authors develop a calculus of nonlinear interactions of Gaussian beams (the WKB interaction calculus), yielding explicit high-frequency expansions for solutions arising from products of beams and enabling analysis of iterated interactions beyond standard FIO methods. By performing higher-order linearizations, they derive integral identities that tie the data to the unknown coefficients, and they demonstrate unique recovery of $F$ when $q_1$ is known. The results extend prior work by handling quadratic nonlinearities without assuming a zero solution and by providing a versatile, beam-based alternative to Fourier integral operator techniques. The framework has potential for broader applications in nonlinear hyperbolic inverse problems and could inform practical imaging scenarios using focused wave interactions.

Abstract

We study the inverse source problem for the semilinear wave equation \[ (\Box_g + q_1)u + q_2 u^2 = F, \] on a globally hyperbolic Lorentzian manifold. We demonstrate that the coefficients $q_1$ and $q_2$, as well as the source term $F$, can be recovered up to a natural gauge symmetry inherent in the problem from local measurements. Furthermore, if $q_1$ is known, we establish the unique recovery of the source $F$, which is in a striking contrast to inverse source problems for linear equations where unique recovery is not possible. Our results also generalize previous works by eliminating the assumption that $u= 0$ is a solution, and by accommodating quadratic nonlinearities. A key contribution is the development of a calculus for nonlinear interactions of Gaussian beams. This framework provides an explicit representation for waves that correspond to sources involving products of two or more Gaussian beams. We anticipate this calculus will serve as a versatile tool in related problems, offering a concrete alternative to Fourier integral operator methods.

Gaussian beam interactions and inverse source problems for nonlinear wave equations

TL;DR

The paper studies inverse source problems for the semilinear wave equation on a globally hyperbolic Lorentzian manifold and proves that the coefficients and the source are recoverable from local measurements up to a natural gauge symmetry. The authors develop a calculus of nonlinear interactions of Gaussian beams (the WKB interaction calculus), yielding explicit high-frequency expansions for solutions arising from products of beams and enabling analysis of iterated interactions beyond standard FIO methods. By performing higher-order linearizations, they derive integral identities that tie the data to the unknown coefficients, and they demonstrate unique recovery of when is known. The results extend prior work by handling quadratic nonlinearities without assuming a zero solution and by providing a versatile, beam-based alternative to Fourier integral operator techniques. The framework has potential for broader applications in nonlinear hyperbolic inverse problems and could inform practical imaging scenarios using focused wave interactions.

Abstract

We study the inverse source problem for the semilinear wave equation on a globally hyperbolic Lorentzian manifold. We demonstrate that the coefficients and , as well as the source term , can be recovered up to a natural gauge symmetry inherent in the problem from local measurements. Furthermore, if is known, we establish the unique recovery of the source , which is in a striking contrast to inverse source problems for linear equations where unique recovery is not possible. Our results also generalize previous works by eliminating the assumption that is a solution, and by accommodating quadratic nonlinearities. A key contribution is the development of a calculus for nonlinear interactions of Gaussian beams. This framework provides an explicit representation for waves that correspond to sources involving products of two or more Gaussian beams. We anticipate this calculus will serve as a versatile tool in related problems, offering a concrete alternative to Fourier integral operator methods.

Paper Structure

This paper contains 18 sections, 34 theorems, 316 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gaussian beam interactions and sketch of the proof
  3. Comparison to FIO calculus
  4. Motivation and related literature
  5. Preliminaries
  6. Gaussian beams
  7. WKB interaction solutions
  8. A Higher order WKB interaction solution
  9. Higher order linearization
  10. Proof of Theorem \ref{['thm_thm1']}
  11. Geometric setup
  12. Choosing special solutions
  13. Uniqueness of $q^2_2$
  14. Uniqueness of $q_2$
  15. Obtaining information about $q_1$ and $F$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $q_1,q_2 , \tilde{q}_1, \tilde{q}_2 \in C^\infty (N)$, with $q_2$ satisfying eq_sign_cond and $F,\tilde{F} \in E^s(N)$. Assume that the corresponding source-to-solution maps $S$ and $\tilde{S}$ are well-defined for $f\in E^s(U)$ small enough. Then implies that gauge_intro hold for $\varphi := \tilde{u}_0 - u_0$, where $u_0$ and $\tilde{u}_0$ solve eq_wave_equation with $f=0$.

Figures (1)

  • Figure 1: On the left is an illustration of the choice of geodesics. The optimal geodesics $\gamma^{(0)}$ and $\gamma^{(1)}$ are indicated by straight lines. On the right is an illustration of the set $W_r$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Definition 2.1: Fermi coordinates
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: Formal Gaussian beam
  • ...and 57 more