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

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

xps=(2/a)*1/Cosh[-a^3*p+a*s-ArcCosh[2/(a*x)]]
ups=s-(2/a)*Tanh[-a^3*p+a*s-ArcCosh[2/(a*x)]]-(2/a)*Tanh[ArcCosh[2/(a*x)]]+u

pmin=20;
pmax=-20;
Plot[xp,{p,pmin,pmax},PlotPoints->100,AspectRatio -> Automatic]
Plot[up,{p,pmin,pmax},AspectRatio -> Automatic]
ParametricPlot[{xp,up},{p,pmin,pmax},AspectRatio -> Automatic]

ParametricPlot[{xps,ups},{s,-1,10}]