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 $u_{\mathrm{xt}}=sinu$ in the quarter-planes $x>0,t>0$ and $x<0,t>0$ , assuming a suitable decay as $x\to +\infty$ or as $x\to -\infty$ . 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" ( $x\ge 0$ ) is uniquely determined by the initial data u(x, 0), $x\ge 0$ alone, whereas for the "left problem" ( $x\le 0$ ), 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).