Resonance fenomena in the nonlinear equation of a proper semiconductor $h^2\Delta u=\operatorname{sh}u$

A boundary value problem of Steklov type for the non-linear semiconductor equation is discussed. Assuming the existence of closed stable geodesic on the surface of a semiconductor the asymptotic solutions which are concentrated in the vicinity of the geodesic are constructed. The solutions are obtained in terms of eigenfunctions if the Laplace operator on a Riemannian manifold and multi-soliton solutions of the Sine-Gordon equation. Similar results are obtained for the mixed boundary value problem.