Asymptotic solutions of Hartree equations concentrated near low-dimensional submanifolds. I. The model with logarithmic singularity

We consider a two-dimensional model Schrödinger equation with logarithmic integral non-linearity. We find asymptotic expansions for its solutions (Airy polarons) that decay exponentially at the “semi-infinity” and oscillate along one direction. These solutions may be regarded as new special functions, which are somewhat similar to the Airy function. We use them to construct global asymptotic solutions of Schrödinger equations with a small parameter and with integral non-linearity of Hartree type.