2026/06/23 by Iryna Karpenko
Mathematics · #math.AP #math-ph #math.MP
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 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) has to be prescribed in addition to the initial data in order to obtain a well-posed problem.