Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics

We study global conservative solutions of the Cauchy problem for the Camassa-Holm equation $u_t-u_{txx}+\kappa u_x+3u{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$ with nonvanishing and distinct spatial asymptotics.