a=1;
u=0.5;
x=0.5;
xs=(2/a)*1/Cosh[a*s-ArcCosh[2/(a*x)]];
us=s-(2/a)*Tanh[a*s-ArcCosh[2/(a*x)]]-(2/a)*Tanh[ArcCosh[2/(a*x)]]+u;
smin=-3;
smax=(1/a)*ArcCosh[2/(a*x)];
gr1=ParametricPlot[{xs,us},{s,smin,smax},
AspectRatio->Automatic,
AxesOrigin->{0,0},
AxesLabel->{"x","u"},
PlotRange->{{-3,3},{-3,3}},
PlotPoints->100]

a=1;
u=0.5;
x=0.5;
xs=(2/a)*1/Cosh[a*s-ArcCosh[2/(a*x)]];
us=s-(2/a)*Tanh[a*s-ArcCosh[2/(a*x)]]-(2/a)*Tanh[ArcCosh[2/(a*x)]]+u;
smin=(1/a)*ArcCosh[2/(a*x)];
smax=10;
gr2=ParametricPlot[{xs,us},{s,smin,smax},
AspectRatio->Automatic,
AxesOrigin->{0,0},
AxesLabel->{"x","u"},
PlotRange->{{-3,3},{-3,3}},
PlotPoints->100,
PlotStyle->Dashing[{0.02,0.02}]]

Show[gr1,gr2]