An integrable generalization of the sine-Gordon equation on the half-line

We analyze a generalization of the sine-Gordon equation in laboratory coordinates on the half-line. Using the Fokas transform method for the analysis of initial-boundary value problems for integrable PDEs, we show that the solution $u(x,t)$ can be constructed from the initial and boundary values via the solution of a $2\times 2$-matrix Riemann-Hilbert problem.