The initial condition for u:

f[x_]:=1/(1+x^2);

The initial condition for u_x is 0.

F[x_]:=f[x]/2;

G[x_]:=f[x]/2;

Snapshot at t=0:

Plot[{F[x]+G[x]},{x,-8,8},PlotRange->{0,1}]

Snapshot at t=1:

Plot[{F[x+1]+G[x-1]},{x,-8,8},PlotRange->{0,1}]

Snapshot at t=2:

Plot[{F[x+2]+G[x-2]},{x,-8,8},PlotRange->{0,1}]

Snapshot at t=3:

Plot[{F[x+3]+G[x-3]},{x,-8,8},PlotRange->{0,1}]

Snapshot at t=4:

Plot[{F[x+4]+G[x-4]},{x,-8,8},PlotRange->{0,1}]

Snapshot at t=5:

Plot[{F[x+5]+G[x-5]},{x,-8,8},PlotRange->{0,1}]

Snapshots at t=0,1,2,3,4,5:

Plot[{F[x]+G[x],F[x+1]+G[x-1],F[x+2]+G[x-2],F[x+3]+G[x-3],F[x+4]+G[x-4],F[x+5]+G[x-5]},{x,-8,8},PlotRange->{0,1}]

Surface plot:

Plot3D[F[x+y]+G[x-y],{x,-8,8},{y,0,5},PlotPoints->40,PlotStyle->{Orange,Specularity[White,20]},Mesh->None,Boxed->True,Axes->True]

An impression of the characteristic lines of u_t+u_x=0:

LineIntegralConvolutionPlot[{{1,1},imagescale},{x,-2,2},{y,-2,2},ColorFunction->GrayLevel,LightingAngle->0,AspectRatio->1,LineIntegralConvolutionScale->2]