Recovery of nonlinearity from the modified scattering map for cubic NLS with a repulsive delta potential

We study the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential and a localized inhomogeneous coefficient.

We prove small-data modified scattering and construct the associated vector-valued modified scattering map, whose two components encode the coupling of the frequencies $\xi$ and $-\xi$ induced by the point interaction.

We show that this map determines both the strength of the delta potential and the inhomogeneous coefficient.

Quantitatively, the delta strength is recovered with Lipschitz stability, while the inhomogeneous coefficients satisfy a Hölder stability estimate.

To our knowledge, these are the first recovery and stability results for a modified scattering map in the presence of an external potential.