The unified transform for evolution equations on the half‐line with time‐periodic boundary conditions

Abstract

Abstract: This paper elaborates on a new approach for solving the generalized Dirichlet‐to‐Neumann map, in the large time limit, for linear evolution PDEs formulated on the half‐line with time‐periodic boundary conditions. First, by employing the unified transform (also known as the Fokas method) it can be shown that the solution becomes time‐periodic for large t . Second, it is shown that the coefficients of the Fourier series of the unknown boundary values can be determined explicitly in terms of the coefficients of the Fourier series of the given boundary data in a very simple, algebraic way. This approach is illustrated for second‐order linear evolution equations and also for linear evolution equations containing spatial derivatives of arbitrary order. The simple and explicit determination of the unknown boundary values is based on the “ Q ‐equation”, which for the linearized nonlinear Schrödinger equation is the linear limit of the quadratic Q ‐equation introduced by Lenells and Fokas [Proc. R. Soc. A, 471, 2015]. Regarding the latter equation, it is also shown here that it provides a very simple, algebraic way for rederiving the remarkable results of Boutet de Monvel, Kotlyarov, and Shepelsky [Int. Math. Res. Not. issue 3, 2009] for the particular boundary condition of a single exponential.