Sine–Gordon theory in a semi-strip

Initial-boundary value problems for sine–Gordon and complex sine–Gordon equations in a semi-strip are treated. The evolution of the Weyl function and a uniqueness result are obtained for the complex sine–Gordon equation. The evolution of the Weyl function as well as an existence result and a procedure to recover solution are given for the sine–Gordon equation. It is shown that for a wide class of examples the solutions of the sine–Gordon equation are unbounded in the quarter-plane.