a=1;
u=0.5;
x=0.5;
xp=(2/a)*1/Cosh[-a^3*p-ArcCosh[2/(a*x)]]
up=-(2/a)*Tanh[-a^3*p+ArcCosh[2/(a*x)]]+(2/a)*Tanh[ArcCosh[2/(a*x)]]+u
pmin=-10;
pmax=-(1/(a^3))*ArcCosh[2/(a*x)]
ppmmax=10
Plot[xp,{p,pmin,ppmmax}]
Plot[up,{p,pmin,ppmmax}]

a=1;
u=0.5;
x=0.5;
xp=(2/a)*1/Cosh[-a^3*p-ArcCosh[2/(a*x)]]
up=-(2/a)*Tanh[-a^3*p+ArcCosh[2/(a*x)]]+(2/a)*Tanh[ArcCosh[2/(a*x)]]+u
pmin=-10;
pmax=-(1/(a^3))*ArcCosh[2/(a*x)]
gr1=ParametricPlot[{xp,up},{p,pmin,pmax},
AspectRatio->Automatic,
AxesOrigin->{0,0},
AxesLabel->{"x","u"},
PlotRange->{{-3,3},{-3,3}},
PlotPoints->100]

a=1;
u=0.5;
x=0.5;
xp=(2/a)*1/Cosh[-a^3*p-ArcCosh[2/(a*x)]]
up=-(2/a)*Tanh[-a^3*p+ArcCosh[2/(a*x)]]+(2/a)*Tanh[ArcCosh[2/(a*x)]]+u
pmin=-(1/(a^3))*ArcCosh[2/(a*x)];
pmax=20;
gr2=ParametricPlot[{xp,up},{p,pmin,pmax},
AspectRatio->Automatic,
AxesOrigin->{0,0},
AxesLabel->{"x","u"},
PlotRange->{{-3,3},{-3,3}},
PlotPoints->100,
PlotStyle->Dashing[{0.02,0.02}]]

Show[gr1,gr2]