Stable inversion of potential in nonlinear wave equations with cubic nonlinearity

TL;DR

The paper addresses stable, local recovery of the lower-order potential $V$ and cubic nonlinearity $h$ in a semilinear wave equation on Minkowski space from limited measurements. It develops a framework based on higher-order (trilinear) linearization and a detailed conormal/symbol calculus for both linear and nonlinear distorted plane waves, deriving explicit principal and lower-order symbol relationships that link data to the unknown coefficients. Central to the approach is the construction of nonlinear conormal waves from three localized sources, the analysis of their cubic interaction $u_{(123)}$, and the definition of a bounded trilinear response operator $T_{V,h}$ whose stability under perturbations yields Hölder-type estimates for $h$ and, subsequently, for $V$ via a truncated light-ray transform. The result is a quantitative, small-data stability theory for simultaneous recovery of the nonlinearity and potential from neighborhood measurements, advancing the understanding of stable inversion for nonlinear hyperbolic PDEs with cubic nonlinearities.

Abstract

This paper investigates inverse potential problems of wave equations with cubic nonlinearity. We develop a methodology for establishing stability estimates for inversion of lower order coefficients. The new ingredients of our approach include trilinear approximations of nonlinear response operators, symbol estimates of distorted plane waves, and lower order symbol calculus.

Figures (3)

• Figure 1: The outer, middle, and inner cylinders respectively represent $\mho$, $\mho_1$ and $\mho_2$.
• Figure 2: The outer, middle, and inner diamonds respectively represent $\mathbb D$, $\mathbb D_1$ and $\mathbb D_2$.
• Figure :

Theorems & Definitions (21)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 3.1
• Lemma 3.1
• Lemma 3.2