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 uₓₜ=sin u u xt = sin u in the quarter-planes $$x> 0, t > 0$$ x > 0 , t > 0 and $$x < 0, t > 0$$ x < 0 , t > 0 , assuming a suitable decay as x→ +∞ x → + ∞ or as x→ -∞ 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 x ≥ 0 ) is uniquely determined by the initial data u ( x , 0), x≥ 0 x ≥ 0 alone, whereas for the “left problem” ( x≤ 0 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).