An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates

We consider the sine-Gordon equation in laboratory coordinates with both $x$ and $t$ in $[0,\infty)$. We assume that $u(x,0)$, $u_t(x,0)$, $u(0,t)$ are given, and that they satisfy $u(x,0) \to 2\pi q$, $u_t(x,0)\to 0$, for large $x$, $u(0,t) \to 2\pi p$ for large $t$, where $q$, $p$ are integers. We also assume that $u_x(x,0)$, $u_t(x,0)$, $u_t(0,t)$, $u(0,t)-2\pi p$, $u(x,0)-2\pi q \in L_2$. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large $t$, shows how the boundary conditions can generate solitons.