The sine-Gordon equation in light-cone coordinates on the half-lines revisited: a Riemann-Hilbert approach.

In this work, we study the initial-boundary value (IBV) problems for the sine-Gordon (sG) equation in the light-cone coordinates u xt = sin u in the quarter-planes x > 0 , t > 0 and x < 0 , t > 0 , assuming a suitable decay as x → + ∞ or as x → - ∞ . Employing the Riemann-Hilbert (RH) problem framework, we demonstrate that these two IBV problems differ significantly with respect to the boundary data required for well-posedness. Specifically, the solution of the "right problem" ( x ≥ 0 ) is uniquely determined by the initial data u(x, 0), x ≥ 0 alone, whereas for the "left problem" ( x ≤ 0 ), the boundary data u(0, t) have to be prescribed in addition to the initial data in order to obtain a well-posed problem. The latter problem is solved using the unified transform method (also known as the Fokas method).