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It has recently been realised that illumination by intensely powerful radiation is not the only path to a nonlinear optical response by a given material. As demonstrated for a layer of indium tin oxide (ITO), strong nonlinear effects can be observed in a material for illuminating fields of quite moderate strength in a neighbourhood of the wavelengths which render it an epsilon-near-zero (ENZ) material. Inspired by these observations we introduce, discuss and analyse a rather different formulation of the governing equations for the Capretti experiment with a view towards robust and highly accurate numerical simulation. By contrast to volumetric algorithms which are greatly disadvantaged for the piecewise homogeneous geometries we consider, surface methods provide optimal performance as they only consider interfacial unknowns. In this contribution, we study an interfacial approach which is based upon Dirichlet–Neumann operators (DNOs). We show that, for a layer of nonlinear Kerr medium, the DNO is not only well-defined, but also analytic with respect to all of its independent variables. Our method of proof is perturbative in nature and suggests several new avenues of investigation, including stable numerical simulation, and how one would include the effects of periodic deformations of the layer interfaces into both theory and numerical simulation of the resulting DNOs. This article is part of the theme issue ‘Analytically grounded full-wave methods for advances in computational electromagnetics’.