We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.
Pelloni, B., & Pinotsis, D. A. (2008). The Klein-Gordon Equation on the Half Line: a Riemann-Hilbert Approach. Journal of Nonlinear Mathematical Physics, 15(Supplement 3), 334-336. https://doi.org/10.2991/jnmp.2008.15.s3.32