  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.
