Sine-Gordon equation on the semi-axis

We investigate the sine-Gordon equation $u_{tt}-u_{xx}+\sin u=0$ on the semi-axis $x>0$. We show that boundary conditions of the forms $u_x(0,t)=c_1\cos(u(0,t)/2)+ c_2\sin(u(0,t)/2)$ and $u(0,t)=c$ are compatible with the Bдcklund transformation. We construct a multisoliton solution satisfying these boundary conditions.