The Fokas Method: The Dirichlet to Neumann Map for the Sine-Gordon Equation

We study initial boundary value problems for the sine‐Gordon equation on the half‐line via the Fokas method, known as an extension of the inverse scattering transform. The method is based on the simultaneous analysis of both parts of the Lax pair and the global algebraic relation that couples known and unknown boundary values. One of most difficult steps of the method is to characterize the unknown boundary values that appear in the spectral functions. We derive the Dirichlet to Neumann map by using the global relation and the asymptotics of the eigenfunctions. Furthermore, employing perturbation expansion, we present an effective characterizations of the unknown boundary value in terms of the given initial and boundary values, and we then derive the first few terms of the expansions of the Neumann boundary value up to the third order.