Construction of solutions and asymptotics for the sine-Gordon equation in the quarter plane

We consider the sine-Gordon equation in laboratory co-ordinates in the quarter plane.The first part of the article considers the construction of solutions via Riemann-Hilbert techniques.In addition to constructing solutions starting from given initial and boundary values, we also construct solutions starting from an independent set of spectral (scattering) data.The Riemann-Hilbert problem, which is the starting point of our discussion is derived by means of the Unified Transform (also known as the Fokas method).The second part of the article establishes asymptotic formulas for the quarter-plane solution u(x, t) as (x, t) → ∞.Assuming that u(x, 0) and u(0, t) approach integer multiples of 2π as x → ∞ and t → ∞, respectively, we show that the asymptotic behaviour is described by four asymptotic sectors.In the first sector (characterized by x/t ≥ 1), the solution approaches a multiple of 2π as x → ∞.In the third sector (characterized by 0 ≤ x/t ≤ 1 and t|x -t| → ∞), the solution asymptotes to a train of solitons superimposed on a radiation background.The second sector (characterized by 0 ≤ x/t ≤ 1 and x/t → 1) is a transition region and the fourth sector (characterized by x/t → 0) is a boundary region.We derive precise asymptotic formulas in all sectors.In particular, we describe the interaction between the asymptotic solitons and the radiation background, and derive a formula for the solution's topological charge.