This　work　is　a　continuation　of　our　previous　work,　in　the　present　paper　we　study　the　mixed　initial-boundary　value　problem　for　general　n　x　n　quasilinear　hyperbolic　systems　of　conservation　laws　with　non-linear　boundary　conditions　in　the　half　space　{(t,x)　vertical　bar　t　＞=　0,　x　＞=　0}.　Under　the　assumption　that　each　characteristic　with　positive　velocity　is　linearly　degenerate,　we　prove　the　existence　and　uniqueness　of　global　weakly　discontinuous　solution　u　=　u(t,　x)　with　small　amplitude,　and　this　solution　possesses　a　global　structure　similar　to　that　of　the　self-similar　solution　u　=　U(x/t)　of　the　corresponding　Riemann　problem.　Some　applications　to　quasilinear　hyperbolic　systems　of　conservation　laws　arising　in　physics　and　other　disciplines,　particularly　to　the　system　describing　the　motion　of　the　relativistic　string　in　Minkowski　space　R1+n,　are　also　given.　(C)　2008　Elsevier　Inc.　All　rights　reserved.