Nonlinearity helps the convergence of the inverse Born series

TL;DR

This paper demonstrates that the inverse Born series can reliably reconstruct nonlinear coefficients in scalar wave equations, including Kerr and polynomial nonlinearities, under small data conditions, with supporting numerical evidence.

Contribution

It extends previous convergence results to include strong Kerr nonlinearities and general polynomial nonlinearities when the linear coefficient is known.

Findings

Inverse Born series converges for strong Kerr nonlinearities with small data.

Convergence results apply to general polynomial nonlinearities.

Numerical examples validate theoretical findings.

Abstract

In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type [8]. We reported conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. In this work, we show that if the coefficient of the linear term is known, an arbitrarily strong Kerr nonlinearity can be reconstructed, for sufficiently small data. Additionally, we show that similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.