Born and inverse Born series for scattering problems with Kerr nonlinearities

TL;DR

This paper develops recursive formulas and convergence conditions for Born and inverse Born series in nonlinear Kerr scattering problems, supported by theoretical proofs and numerical experiments.

Contribution

It introduces new recursive formulas and convergence criteria for Born series in Kerr nonlinear scattering, with rigorous proofs and numerical validation.

Findings

Derived recursive formulas for Born series operators.

Proved boundedness and convergence conditions for the series.

Validated results through numerical experiments.

Abstract

We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.